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A
ag / as / le / chuig / i / go = N "H"
X DENTLS ∆ + "an" > "h" ex. leis an bhothar
right, except the missing ó (bhóthar) (but in Ulster we rather say "leis an bhealach (mhór)"

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Examples:
I think she was saying no lenition except with the article "an" and of course no lenition with DNTLS.
Use of an "H" when the noun following begins with a vowel like "go hÉireann".
yeah you prefix h to vowels
(be careful, it is "go hÉirinn", not *go hÉireann, because "Éireann" is the genitive case, but "go" requires the dative : Poblacht na hÉireann but tá mé ag gabháil go hÉirinn). (and Éirinn and Éireann aren't pronounced the same way: Éirinn is AY-rhiñ (with the Spanish sound ñ, ny in English) while Éireann is AY-rhunn.
I've recorded them:
http://fr.forvo.com/word/%C3%A9ireann/#ga http://fr.forvo.com/word/%C3%A9irinn/#gaQuote:
ag - ag geata / ag an gheata
yes
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as - as cólaiste / as an chólaiste
yes except the síneadh fada is on the a

coláiste
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le - le cúnamh Dé / leis an chúnamh
yes
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chuig - chuig bord / chuig an chólaiste
yes (but "coláiste" again)
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i - i mbíalann / sa bhíalann
yes (but no síneadh fada in that word, actually "ía" doesn't exist in Modern Irish, it's always "ia")
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go - go Méiricea ?
yes: go Meiriceá (I think we say go Meiri
ce in Donegal)
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B
faoi / gan / ó / roimh = + "h"
Is there seimhiú without a definite article and an urú when using the definite article?
after faoi, ó, roimh : séimhiú with article (in the singular) and without articles.
"Gan" is a bit different because it doesn't lenite everything (see New Irish Grammar p.14 - if you have it)
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faoi - faoi gheasa / faoin gcéad *
normally it's "faoin chéad" in Donegal. (using urú after "faoin" is a Connachta/Munster thing).
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gan - gan choinne
gan mhaith
gan dóchas (seems it sometimes lenits and sometimes it seems to not lenit)
Yeah, "gan" lenites, except...
- d, t, f, s
- qualified nouns (gan bréag ar bith, but if you remove "ar bith" it becomes "gan bhréag")
- verbal nouns when it means "not to+verb" (I told him not to marry = dúirt mé leis gan pósadh)
- proper names (gan Micheál)
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ó - ó chíon is ó chóngar
yes
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ó Thuaidh
the ó that is used to express directions is a completely different word ; yes "thuaidh" is lenited but it's always lenited actually

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C
ar + "h" ar chapall... leis eisteachtaí
yes, ar chapall. And yes there are exceptions, mainly set phrases.
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I'm supposing my notes mean "ar" usually seimhiús, but is followed by urú when a definite article is used.
ar - ar bhothar / ar an mbothar
ar capall / ar an gcapall
"ar an" is followed by urú only in Munster and Connachta, as "faoin" etc. In Ulster it's much simpler, it's always séimhiú: ar an bhóthar (or ar an bhealach), ar an chapall... and ar chapall.
Another thing I see from what you wrote, looks like you were taught stuff like
ar + "h". This is very likely to confuse learners, because what is "h"? Is it prefixed h (as in "go hÉirinn") ? or séimhiú? Learners can't know. Because prefixing h- to vowels and séimhiú on consonants never happen in the same situations : either the preposition prefixes h to vowels and don't lenite consonants, or it lenites consonants and then it doesn't change anything to vowels.
I know many Irish teachers say "that preposition adds "h" " and that's why learners often don't know if it means they need to add h- to vowels or lenite consonants and then make mistakes...
It's much clearer to teach for example:
le prefixes h to vowels
faoi lenites consonants (= faoi triggers séimhiú, = faoi aspirates).