I have hard time parsing and understanding how typical questions about the direct object in progressive constructions are grammatical and came to be (except for the first one in the topic name, which is clear to me).

To say ‘I am doing it’ one says táim á dhéanamh, lit. ‘I am at its doing’ (or ‘to its doing’, since ag and do got mixed in such constructions). Another way, afaik common in Munster, is to make it passive ‘it is being done by me’: tá sé á dhéanamh agam, lit. ‘it is at its doing by me’.

‘I am reading the book’: táim ag léamh an leabhair or tá an leabhar á léamh agam.

The second one is easy to turn into question, since we ask for the grammatical subject. We get a very straightforward direct relative clause: cad ’tá á dhéanamh agat? (cad atá á dhéanamh agat?) ‘what is being done by you’, lit. ‘what’s-it that’s at its doing by you?’. This seems to be the only way to ask such a question that Doyle’s and Gussmann’s An Ghaeilge teaches and I believe this is the most common way in Munster.

But the first one is active, the action is described by the verbal noun which takes its object in genitive (or as a possessive pronoun), and thus the sentence syntactically does not have any direct object, it is ‘I am at something’s doing’, the verb is ‘am’ and it is not transitive, it doesn’t take an object. The predicate is ‘at its doing’, a prepositional phrase. Thus, to turn it into a question, I would naïvely use an idirect relative and ask ‘what’s-it at whose doing you are?’:

*céard a bhfuil tú á dhéanamh?, or *cad go bhfuileann tú á dhéanamh?, or the like

but I think nobody actually speaks like that. I wouldn’t be surprised that much by eg. *céard atá tú á dhéanamh? – as I think (though I might be wrong) the indirect relative form in affirmative clauses in some areas is being replaced by direct one, similarly to how in Scottish, afaik, the same form is used for both direct and indirect ((a) tha thu rather than *a’ bheil thu, eg. an gille a tha a mhàthair bochd vs. an buachaill a bhfuil a mháthair tinn/bocht) – but that’s also not used IMO.

What I can find is that the actual question, outside of Munster (and sometimes in Munster too), asked is something like

céard atá tú a dhéanamh?, caide ’tá tú a dhéanamh?, cad ’tá tú a dhéanamh?

and I don’t know how to understand those – how do they fit into the Irish syntax? What is a dhéanamh here?
Does it come from do dhéanamh like in rud do dhéanamh, rud a dhéanamh ‘to do a thing’? If so, the question seems to ask something like ‘what are you to do?’ and turned into a positive sentence would be *táim (do?) é a dhéanamh which makes no sense, but I think I can see how such questions could evolve from something like táim chun é a dhéanamh – but that has a very different meaning.
Or is it just a plain possessive a ‘its’? But then it’s even more unclear to me, is the question literally ‘what are you its doing’ with a hidden copula (and why a direct relative then)?

How are they supposed to anticipate táim á dhéanamh, táim ag léamh an leabhair, etc. as the answer?
And how old are they – would eg. Keating use them, or are they a newer innovation?

Also sometimes I see things like *cad atá tú ag déanamh. A few results from Google suggest it’s frowned upon by some poeple, which makes sense as it indicates very Urban-Irishy ungrammatical *táim ag déanamh é. But on the other hand that’d be a pretty straightforward to explain construction (since it’s awkward to turn táim á dhéanamh directly into question, the ag + verbal noun only in questions would be allowed to take a direct object, and again afaik Scottish Gaelic does exactly that, eg. dè tha thu a’ dèanamh? vs tha mi ga dhèanamh) – do native speakers use it, or is it just an Urban Irish thing?

Yes, verbatim this means something like ‘what are you to do?’

The object of a verbal noun in active progressive constructions is in genitive relation and the progressive particle is "ag".
Except in cases the object can't be in genitive relation. In all those cases, a (< do) replaces ag:

1) relative clauses with object as antecedent: an rud atá mé a dhéanamh (the thing that I’m doing)
2) object questions (similar to 1)): Céard atá mé a dhéanamh? (What am I doing?)
3) negative sentences, object fronted: Rud ar bith níl mé a dhéanamh. (Nothing I’m doing.)
4) seminegative sentences, object governed by ach: Níl mé a dhéanamh ach an rud seo. (I’m only doing this thing)

You can use á and an indirect relative clause in 1) and 2) instead of a and direct relative. But this is rare.
1) an rud a bhfuil mé á dhéanamh ("the thing at whose doing I am")
2) Céard a bhfuil mé á dhéanamh?


Because of different word order.

GRMA!



So basically this is analogical to the Scottish dè (a) tha thu a’ dèanamh? with the only difference being that Scottish keeps ag (a’) here, and Irish changes it to leniting do/a? That is surprisingly similar to (Connacht, standard) Irish using do in affirmative progressive with pronouns, tá sé do mo bhualadh, do bhur mbualadh while Scottish keeps ag, tha e gam bhualadh, gur bualadh.

Is it true for all dialects (at least those that don’t require – if requiring it dialects exist – the use of passive an rud atá á dhéanamh agam, cad atá á dhéanamh agam, etc)? Are there any that use ag here (cad atá tú ag déanamh?, an rud atáim ag déanamh)?

I'd think it is true for all dialects (or passive is used). "Ag" doesn't occur.

I think "caidé atá tú ag déanamh" is common in Donegal. It's also what they say in Scotland.
So I wouldn't say it's wrong.

_________________
Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

In Tobar na Gaedhilge there are only examples with a in the Ulster section:
Caidé atá tú ’fhoghluim anois?
Caidé tá sé a dhéanamh?
Caidé tá mé a rádh leat?
Caidé tá tú a iarraidh?
Goidé tá tú a léigheadh?
Goidé tá tú dh’iarraidh?

The same in Corpas Nua with Ulster native speakers:
caidé atá mé a mhaíomh?
Dheaidí, caidé atá tú 'dhéanamh?
Goidé atá sibh a iarraidh?
Goidé atá tú a rá?
Níl a fhios ag aon duine aon lá sa bhliain goidé atá Dia a dhéanamh.

Examples with ag are subject questions or double progressives
Caidé tá ag caitheamh ort?
Caidé tá ’teacht ort?
Goidé tá tú ag brath a dhéanamh?

Might depend on the parish, and for sure, the literary language is more conservative.
I read "caidé atá tú ag déanamh?" etc in dialectology books, at least in Deilbhíocht Ghaeilge Ghaoth Dobhair and I guess in a few other books.

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Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

Since do here supposedly comes from confusion with other verbal noun constructions involving do but originally used ag ← oc, I’d risk arguing that the literary language here is more innovative and using ag is more conservative. :P

Same. I've heard it a lot in Conamara too. Reminds me of an anecdote I read:

"Scoláire Gaeilge agus Gaeilgeoir ón gcliabhán a bhuail isteach i rang chugam lá amháin, chuir sé iontas mo sháith orm. Cigire scoile a bhí ann. Bhreathnaigh sé ar charnán aistí a bhí mé tar éis a cheartú. "Sé an locht amháin atá ort," a deir sé, "go bhfuil tú ag cur an iomarca trioblóide ort féin leis na haistí seo. Seo rud, cuir i gcás, atá ceartuithe agat: "an rud a bhí sé ag déanamh." Ba cheart a fhágáil mar atá sé agus gan "an rud a bhí sé a dhéanamh" a chur ann mar a rinne tusa. Is fás nádúrtha é atá ag teacht ar an teanga. Tá sé le clos go coitinn i gConaimeara."

"Níl tú á rá liom?" arsa mise. "Ach cá bhfuil an áit sin - Conaimeara? Is áit é nár chuala mise trácht ariamh air, agus is beag contae sa tír nár fhág mé lorg mo choise ann. Más é Conamara atá i gceist agat cuirfidh mé céad punt leat (ní raibh céad pingin agam), nár chuala tú ag aon duine ansin ariamh é ach ag duine eicín ar mhúin tusa nó do leithéid eile dhó é ar scoil." (Chaith sé scathamh maith ina oide scoile sular toghadh ina chigire é). Níor chuir sé an geall liom, rud a d'fhág beo bocht mé ariamh ó shin. Bhog leis chuig rang eile agus níor chuir an seanchas ní b'fhaide."

-Seán Ó Ruadháin, Ceart nó Mícheart

Is cosúil go bhfuil sé ina chnámh spairne ag lucht gramadaí le fada an lá!

It's funny you mention this anecdote, because I've been told opposite ways from two native speakers, both of whom teach the language. One, from Ros Muc, said that it was absolutely "a dhéanamh", whereas the other, from Leitir Mealláin (and younger), said "ag déanamh" was correct. It was interesting to see that.