Small typo, but we don't want to mislead anyone:

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.

And is the pronunciation distinct from the colloquial a'am [= agam]?

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.

For some reason the links to these are dead both from the forum and from my bookmarks.

seomra codlata

Séo é mo sheomra codlata.

Does the h here have to do with the article and seomra codlata? There is no article, definite or not, just a possessive, ("my"). But, in other words, is seomra a masculine noun like in the wombat's link and you only see the h when its in a sentence?

mo causes séimhiú on both masculine and feminine words that start with p, t, c, f, s (usually), b, d, g, and m.

In one of the examples in RS, a boy is introducing himself to a girl. He says:
"Dia duit. Tá sé deas bualadh leat."
This I read to say, "God to you. Is it nice meet with you."

In another example, one I posted before, a man is introducing himself to another man. He says:
"Dia duit. Is deas liom bualadh leat."
This I read to say (as per an instruction from someone earlier), "God to you. Is nice with me meet with you."

My question is, why the difference in the greetings? Is it just like we say in English "Nice to meet you" v. "Pleasure to meet you?" I.E. just two examples of saying hi. Or does the type of greeting depend on the greeters, i.e., boy and girl, man and man?

They both basically mean, 'Hello, nice to meet you'. Gender of speaker or other person is irrelevant.

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As I understand it, certain prepositions cause lenition, while only the preposition "in" causes eclipses.

So if cathaoir is chair, why is "on the chair" gcathaoir and not chathaoir? This seems like lenition and not eclipses.

Wait...is it because it's "on the"? I.E. ar an, which triggers eclipses? preposition + article in the singular = eclipses?

Yes, i causes eclipsis on its own. Most other prepositions cause lenition on their own, except ag, as, chuig, and le.

However, when there is an intervening article an, eclipsis is the rule in Connemara/Connacht and Munster (lenition in Donegal/Ulster), except for sa = i + an which causes lenition in CO, Munster and Donegal/Ulster, but eclipsis in Connemara/Connacht.

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WARNING: Intermediate speaker - await further opinions, corrections and adjustments before acting on my advice.
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

Well, I know in Munster people say "am" as /@m/, at least that is what I have heard. Maybe a few say /am/.

Forms of ag and possessive pronouns use the fada to indicate stress, which is uncommon today.

ag á = at his

With the fada indicating stress, not length.

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal