An Carghas is Lent in Irish.

How would you say, 'Did you give up anything for Lent?' Anything that I can came up with for the 'to give up' sounded like Béarlachas. So I went to the dictionary...

'troscadh an carghais a dhéanamh' and 'an carghas a dhéanamh' are in De Bhaldraithe.

Are these correctly used, 'An ndearna tú troscadh an carghais?' and 'An ndearna tú an carghas?'

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I would say " Ag déanamh An Carghas" if you were giving up something for Lent.
Well actually I'd say "ag déanamh An Charghas" but other spellings seems to be the Standard.
"troscadh agus tréanas" -
Troscadh only applies to not eating food as far as I know - fasting.
Tréanas is not eating meat. Like on Ash Wednesday and Good Friday.

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It is recommended that you always wait for three to agree on a translation.
I speak Connemara Irish, and my input will often reflect that.
I will do an mp3 file on request for short translations.
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I'd say you're right Bríd about ag déanamh An Charghas (Genitive).

FGB gives Rinne mé an Carghas ar an ól - I abstained from drink during Lent.

Tá an tuiseal ginideach i ngach uile áit! Níl é cinnte anois go bhfuil mé ag iarraidh bheith cairdiúil leis.

It must be 'leis' and not 'leí'; nothing that frustrating could be female.

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Could someone please explain what triggers the use of the genitive here?

GRMA

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Táim ag foghlaim Gaelainn na Mumhan

Tá fáilte roim nach aon cheartú!
I am a learner. Any translations offered are practice and should not be used unless confirmed.

The verbal noun triggers the genitive.

Peil---Ag imirt peile
Ceol---An seinm ceoil
An ghrian---Ag sú na gréine

Aw, crumb! I think I knew that. For some reason I don't see "noun" when I read verbal nouns, I see "verb" so I need more work with those.

GRMA as an gcabhair.

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Táim ag foghlaim Gaelainn na Mumhan

Tá fáilte roim nach aon cheartú!
I am a learner. Any translations offered are practice and should not be used unless confirmed.