Thanks for answering my question.
It only took 3 years.

Well, at the time of your asking the question all Irish I had was Dia duit and póg mo thóin (oh, and go n-ithe an cat thú is go n-ithe an diabhail an cat), so I’m afraid I couldn’t help back then. ;-)

Anyway, this topic and the Séadna one are very helpful, I will probably post from time to time my own questions about grammar in them, as I am now progressing in Séadna (so perhaps in that topic more, not this one).

Technically what is going on here is "delenition", "am" causes the lenited form to return to the unlenited form. However sometimes the delenition isn't back to the same letter, hence you would see:

am' báthair

for "at my mother".

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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

Wow, didn’t see that coming. But it makes sense, especially since nasalization is lost in speech after séimhiú. Looks a bit like some nua-urú, at this moment I wouldn’t be surprised if I saw it written like am bmháthair… :P

What happens to f in this case? Would it be, as I wrote, am’ feiscint, or rather am’ ’eiscint (ie. would the deletion of a sound be undone)?

It's rather a kind of dissimilation: m-m -> m-b.
Urú is a form of assimilation: -n/m-b- -> -m-

m-mh (or bh) /m-v/ to m-b /m-b/ (eg. am’ mháthair to am báthair).

Change from bilabial nasal – fricative to bilabial nasal – plosive (or bilabial nasal – lenis → bilabial nasal – fortis). Not sure if I’d call it assimilation, but certainly it’s not dissimilation either. But you’re right, it does not quite work like eclipsis which might change fortis sound (devoiced) to a lenis one (voiced), besides nasalization.

Still interested what happens with /f/ that should be bilabial (but not necessarily is in modern speech) and disappears completely when lenited.

I think it's worth pointing out that this applies to dhá only.
His two houses: a dhá thigh
Her two houses: a dhá tigh
Their two houses: a dhá dtigh

But numbers 3 and over are handled differently, with the numeral lenited or eclipsed. If this is different in a dialect you know, please let us know. These are the forms I know:

His three houses: a thrí tithe
Her three houses: a trí tithe
Their three houses: a dtrí tithe

Of course, not all dialects use the plural after numerals over 3, and so that comes into play somewhere in this.

My research leads me to believe:
With 4 houses it is: a cheithre tithe, a cheithre tithe and a gceithre tithe. i.e that cheithre does not delenite to cheithre even if it means "her four houses".
With 5 houses it is: a chúig tithe, a chúig tithe agus a gcúig tithe, i.e. that chúig does not delenite to cúig even if it means "her five houses".
With 6 houses: a shé tithe, a sé tithe, a sé tithe
With 7 houses: a sheacht tithe, a seacht tithe, a seacht tithe
With 8 houses: a ocht (d)tithe, a hocht (d)tithe, a n-ocht (d)tithe
With 9 houses: a naoi dtithe, a naoi dtithe, a naoi dtithe
With 10 houses: a dheich tithe, a deich tithe, a ndeich tithe

However, not every combination is attested in the Irish of Ua Laoghaire or Ó Loingsigh, so some gaps are having to filled in by guesswork....