"How's about ye?" is one version of a greeting that you might hear in Northern Ireland.
focloir.ie translates the slightly more polished "what are you about?" as cad é atá fút? or cad atá ar siúl agat? (this may be an example of Hiberno-English - though whether the cad é atá fút? is a literal translation of the English, or "How's about ye?" is derived from the Irish, I have no idea).
Since the a in a dhéanamh is a preposition, I took it as being extraneous following the preposition faoi, similar to Tá fút teacht shown in use #4 here. I could certainly be mistaken about its extraneousness in a question, though. Cad requires a relative clause, but the direct relative particle a is included in atá.
So, I asked the question to someone more knowledgeable than me, and I received this reply:
'Sé "Céard atá faoi a dhéanamh" an leagan ceart dar liomsa. Is féidir an "a" a bheith báite i gcaint tapaidh nádurtha, "elision" a tugtar air i mBéarla, agus dá bhrí sin chloisfeá "Céard atá faoi dhéanamh" ó am go ham. Níor chuala abairt den tsaghas "atá faoi déanamh" riamh.
Scilling, having trawled through Dinneen's dictionary to find obscure words, guessing at what classical Irish word (Dinneen uses Classical Irish spelling) could have produced the dialect form I had found in an old text, you have given me the fantastic experience of somebody resorting to Dinneen to understand me! :)
Indeed. If you can find a copy of Learning Irish (it's on the internet), on page 68 it gives examples like Cé atá Cáit a phósadh for "Who(m) is Cáit marrying?" and talking with native speakers, they would say things like Céard atá tú a dhéanamh instead of Céard atá tú ag déanamh. I'm basing it off those constructions, and how it becomes a + VN if the object is before ("Cáit is marrying who?" "You are doing what?")
Edit: It's also possible that these structures only occur in the progressive. I'm extrapolating, but have yet to find anything definitive. I believe it will be the same because the direct object of the VN, the céard/cad still occurs before the VN.
Yes, I know. It's not because of the relative clause. It's because of cad, which is the object of déanamh. It's similar to Céard atá tú a dhéanamh, where, because 'what' is the thing being done, you just he a + lenition structure. Though I could be wrong too. Just basing in other structures with the VN and a question.
Generally, "seo" would be used with "here" ("anseo"), and "sin" with "there" ("ansin"). But I think that in the phrase "what's under there?", "there" is being used as though it was a pronoun, replacing "the chair" or "the car" in a sentence like "what's under the chair?", "what's under the car?". In that case, you could just say "Cad atá faoi?" to a person who was already looking or about to look under the chair/car, and it would be understood as "what's under there?".
If you're actually pointing at a specific point you might say "cad atá faoi sin?", but you might need to be more specific and say "cad atá thíos faoi sin?" to make it clear that you were talking about "underneath", because "faoi seo" and "faoi sin" are usually used in phrases where they mean "about this" and "about that" - "an bhfuil tuairim agat faoi sin?" - "do you have an opinion about that?"
While it is is common to elide the a in atá in questions (Conas tá tú? being a very common example), that's not happening here. If she were saying cad tá faoi? there wouldn't be a gap between cad and tá when spoken - cad would run straight into tá (individual words aren't automatically separated in spoken Irish, unless there is a reason for such separation). What you are interpreting as a gap between the two words is just an unstressed schwa, the a in atá, because the stress in atá is on the accented letter.
Similarly, the á is temporally "short", but phonologically "long" - she moves onto the "f" in faoi quickly, but the sound of the á is well within the normal range for á, a sound that varies considerably between dialects.
You really aren't doing yourself any favours by over-analyzing these sounds in this way, because normal speech is extremely variable, and we achieve comprehension by ignoring or smoothing over those variations. But you're deliberately picking out those variations, and focusing on them in an unhelpful way, misinterpreting the variation as evidence of mistakes, rather than as evidence of normal variation.