I could not spot anything on first reading!

Hmm... which mature source contains the plural erch for orch? It is also included in the Wortliste Sindarin-Deutsch at sindarin.de, though it’s not in WJ:390, Letters:178 or Lambenórë’s index of PE17 words.

“Erch” is from Noldorin, from what I can tell. I included it because there are examples of the same pattern in mature Sindarin. It also matches the “lost final-syllable vowel” so it makes sense that it’d be one such word. But, for Woodelves, learning Sindarin as a second language, with their accent that Frodo found difficult to understand, I could see them not knowing about the lost vowel, and treating the remaining vowel as a final-syllable vowel. So, they’d say the plural “yrch”.... that’s one of the complexities I’m trying to get across in the lesson: the differences of dialect. 

Regarding erch:

The plurals erch, eirch are from LR:406, still Noldorin. However, the pattern itself recurs in Sindarin, see egl, eigl from ogl (PE17:149), for example, so I wouldn’t dismiss them. Still, I would prefer yrch because it is clearly mentioned in the specifically Sindarin sources: The Lord of the Rings, Quendi & Eldar and some places in PE17.
At one point in PE17 Tolkien toys with the idea that yrch should be a Nandorin dialectal form (as basically does with any words and names potentially related to the Nandor), but other sources have no mention of it. Moreover, the pattern o > y is amply attested otherwise.

Regarding the lesson:

The development of long vowels should be like that:
CE /e:/ > OS /i:/ > S /i:/
CE /o:/ > OS /u:/ > S /u:/
CE /a:/ (perhaps rather /ɑ:/) > OS /ɔ:/ > S /au/ (North Sindarin retains /ɔ:/)
Short /a/, /ε/, /i/, /u/ are unchanged. Unless I’m oblivious to some recently published material, there shouldn’t be any short close /e/, /o/; or they’re not relevant.

The change /œi/ > /y/ doesn’t seem to be right, /œ/ simply loses rounding in Sindarin, giving /ε/, so /œi/ should give /εi/ (but not /ai/, I would think).
The mutation of /ɔ/ to /y/ is most likely caused by a different intermediate change, namely raising to /u/ (see Bertrand Bellet’s excellent article on vowel mutations, for example). Hence: *coroni > *coruni > *ceryn.

It’s somehow confusing to see geil → gail, celeir → celair in a textbox, as they look like plural formations, although they are shifts in the singular.
Also, it is not right to say that when ”/i:/ occurred in a single-syllable word, it became EI”. Long /i:/ was unchanged. Geil  and gail come from giljā with a-mutation causing the lowering /i/ > /ε/ and subsequent i-mutation. The plural was *gilī, yielding gîl.
In fact, one can formulate this as a simple rule: Whenever there are two variants with eI and ai, the plural has î. When there is only ai without a variant, the plural (most likely) stays ai.

Fang ‘beard’ is from sphanga (so attested in Old Noldorin, CE spangā). SPÁNAG- is the root, with the acute denoting stress.

I’m not saying that orch → yrch isn’t valid, I’m saying that it is also valid. Using final-syllable mutation on previously non-final syllable vowels I think could be used to mark the speaker as a young elf, or an elf that learned Sindarin as a second language, or a Gondorian using it as a second language.  That would be the majority of the Sindarin speakers making the plural of orch → yrch.

For /a:/ (or /ɑ:/) I think that the /au/ comes out only when on a stressed syllable, and then, only sometimes.
Anor vs naur vs nórui vs bauglir
Glorfindel vs Glaurung
I think it suited Tolkien’s tastes to have scary/evil names and words with the AU diphthong in them.

/u/ → /ɔ/ and /i/ → /ɛ/ could happen because of A-affection... I forgot /i/ in the chart! I’ll go fix that now...

 /œi/ → /y/ ... Do we have examples in Sindarin of /œi/ becoming /ei/? A more detailed breakdown of this transformation I saw something like this:
 /œi/ → /yi/ Vowel raising from partial assimilation
/yi/ → /y/ Vowel deletion

So this theory, the /o/ is raised by the /-i/ before being pulled to the front, but this only happened to the /o/ closest to the /-i/?

I like how you put that AI/EI rule, it’s very clear.

squints at copy of etymologies That could indeed be an acute accent... my copy is rather worn-out. I’ve been meaning to replace it for years now, but I’m so cheap I just keep taping the pages back in...

The revised lesson is up, what do you think?

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Using final-syllable mutation on previously non-final syllable vowels
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I’m confused.. o > y is the final-syllable rule, o > e the non-final one...

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I think could be used to mark the speaker as a young elf, or an elf that learned Sindarin as a second language, or a Gondorian using it as a second language.
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Yes, one important thing to realize is that Sindarin is perfectly happy with several plural forms existing at the same time, just as, say, English can live with several past participles like ‘proved/proven’ or ‘showed/shown’.
Unfortunately, Tolkien never gives any explanation on who uses which one. Age might play a role, but just as well social status, region or simply personal preference - it’s all pure conjecture from here on.
In fact, I don’t think we can say that yrch would be the newer form. Letter #168 mentions that enyd from onod was the ‘correct’ plural, ened might be used in Gondor; so one could argue that the application of the non-final rule to final syllables in a later innovation, thereby making erch the younger one. But they may have just as well developed in parallel.

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/œi/ → /y/ ... Do we have examples in Sindarin of /œi/ becoming /ei/?
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The only examples of /œi/ I’m aware of are actually from Noldorin:
plurals:
gœlœidh, geleidh
dœrœin, deren
non-plurals:
fœir, feir
rhœin, rhein and tellœin, tellen
The further shortening of the diphthong to e in _ tellen, deren_ must be due to final unstressed position, as also seen in _ athae, athe_, for example.

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So this theory, the /o/ is raised by the /-i/ before being pulled to the front, but this only happened to the /o/ closest to the /-i/?
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Exactly, and there is good evidence for that, namely the development of Osse’s Valarin name Oššai in WJ:400: Ossai > ussi > yssɪ̯; and also the Early Noldorin plural of orn ‘tree’: ornei > urnī > yrn (PE13:116).

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I just keep taping the pages back in...
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Mine are also falling out and are quite battered at the edges.. I recall the mention of someone having published the Etymologies as a booklet without the Lost Road part. Now that is a great idea!..

I’m confused.. o > y is the final-syllable rule, o > e the non-final one...

That’s the point... there’s a small number of plurals in Sindarin and Noldorin that have I-Affection only, and no final-syllable vowel mutation. Hypothetically, there was a final syllable that was lost. In some words, that lost final syllable is retained in the plural form (ôl>ely), but in others, it is also lost (fang>feng). Of course, such special forms could easily be lost by analogical changes, and the different forms could stay around simultaneously, as you said:

Yes, one important thing to realize is that Sindarin is perfectly happy with several plural forms existing at the same time,

Which is the point of introducing these cases. When I translate into Sindarin, I try to keep it from a perspective of a character that’s living there. I very rarely translate as from my own perspective. When I do, I chose the dialect that gets out what I want to say the easiest, usually the Noldor-influenced Western Sindarin because I can Sindarinize Quenya words to fill in vocabulary, or the Doriathrin dialect if I’m feeling fancy.

It’s true that we have very little information about what the features of various dialects are. These are some differences that I’ve inferred are likely. I didn’t assign them at random... though I do cherry-pick from Tolkien’s many revisions to some extent, especially when it comes to the Woodelven dialect. That one is probably the most iffy of my hypotheticals piled on hypotheticals.

Exactly, and there is good evidence for that, namely the development of Osse’s Valarin name Oššai in WJ:400: Ossai > ussi > yssɪ̯; and also the Early Noldorin plural of orn ‘tree’: ornei > urnī > yrn (PE13:116).

Thanks! I shall include this in the main chart. Do you think that the raising happened to the /e/ and /ɑ/ too? It could explain why there seems to be two different E’s in Old Sindarin at one point.

BTW, what do you think the history of “it rains” would look like? In Noldorin it’s œil, but I think it would be yl or ŷl in Sindarin. What do you think?

Mine are also falling out and are quite battered at the edges.. I recall the mention of someone having published the Etymologies as a booklet without the Lost Road part. Now that is a great idea!..

What I need to do is get a hard-cover copy, so I can use the page number citations in Dragonflame. Right now, the citations are useless to me, other than giving very vague ideas of where I could find a word. Also, I bet it would last longer. I tend to annotate books as I read them, to help me process the information, which means they get worn out more quickly. I met up with Aaron Shaw (we grew up in the same valley in Montana, barely 10 miles from eachother, only a couple years apart in age), and his Tolkien books are pristine in comparison to mine.

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 there’s a small number of plurals in Sindarin and Noldorin that have I-Affection only, and no final-syllable vowel mutation. Hypothetically, there was a final syllable that was lost. In some words, that lost final syllable is retained in the plural form (ôl>ely), but in others, it is also lost (fang>feng). Of course, such special forms could easily be lost by analogical changes, and the different forms could stay around 
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We don’t really need to hypothesize, usually we know what happened: ôl has an additional syllable in the plural because the Old SIndarin forms were *olos, pl. *olosi. The latter gave *olohi by lenition and ohi eventually contracted to *oi _> _y.
On the other hand, OS *fanga had the plural *fangī  (giving fang, feng respectively) since final -āi shortened to -ī by the Old Sindarin/Noldorin stage. The youngest example of this is found in PE21:58 with _ góndōi > gondoi > ON gondī_.

As to why we get feng instead of *faing, in this case it might be due to the velar nasal (same with ranc, renc), perhaps there was actually an intermediate form *faing.
But in general one simply has to accept there were competing sound changes at work in Sindarin which gave rise to alternate forms. Perhaps words were widely loaned back and forth from various dialects - that sort of thing happens in natural languages as well.

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Thanks! I shall include this in the main chart. Do you think that the raising happened to the /e/ and /ɑ/ too? It could explain why there seems to be two different E’s in Old Sindarin at one point.
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The raising must have happened to both mid vowels o and e, giving u and i respectively (at no point does one need two different e’s). The latter gave the brethel, pl. brethil pattern.  If the vowel a was also affected, I don’t think the result would be distinguishable from later i-mutation.

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BTW, what do you think the history of “it rains” would look like? In Noldorin it’s œil, but I think it would be yl or ŷl in Sindarin. What do you think?
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I suggested a recipe a while ago (see the last paragraph here: http://sindanoorie.net/art/ei_ai.html#sind_disc) according to which it would be either *eil or *uil.

Connecting ulyā with phoryā is promising, I think. I’ll have to mention the uncertainty with the Sindarin-ization of this Noldorin term, though.

As for eil, it wouldn’t matter where the EI came from, if it was in the final syllable, I imagine that it would be swept into AI.