What dialects use these constructions (example: an teach s'againne). I know for a fact they're used in Donegal, but do they appear much in Connacht or Munster? Also, is there any extra connotation behind them rather than another way to do possession-like things. Like, is there a difference between an teach s-againne and ár dteach?

Yes we say "teach s-againne" etc in Conamara too. Pronounce something like "shan(g)a".

I'm not able to explain the difference in usage between it and "ár dteach". Sorry.

There's no difference in meaning per se, it's just a historical difference in perspective.

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There's also the possessive formed with "le"...for example "A Dhia, a Dhia liom!" (O God, my God!). In "The Secret of Roan Inish" one group refers to their house as "Teach linn."

Redwolf

s’againne (or s’ainne as Bríd said), s'agaibhse are used in Ulster & Connacht
s’agamsa, s’agatsa are confined to Ulster. There's seo ’amsa, seo ’atsa in Connacht instead.
In Munster you have to use seo agamsa, seo agatsa, etc.