What would these be in Irish ?

- a counting-out or eliminating game - a counting rhyme is - rann comhairimh/cuntais, but I'm not sure that that would be the same thing.

When they're playing a chasing game (cluiche tóraíochta) what do you call the place where you are 'safe' or 'home', and what do you say when you get there ? In English it's flixie and you cross your fingers, though this probably varies from playground to playground. In French it's pouce/thumb.

Interesting. I've never heard of "flixie" or crossing one's fingers for that reason in North America. The safe place is usually just called "home". Here , crossing one's fingers is done for luck, or (if done in a hidden way) theoretically excuses someone who is telling a "fib" (a small untruth).

Some of the calls in our children's games are German in origin, having come from the many German-speaking immigrants (nearly one-third of Americans have at least some German ancestry), although many people even in North America are not aware of that. For example, when a chasing game (which we usually call "tag") is over, a common call is some variation on "olly-olly oxen free" or "olly-olly in come free", meaning that it's safe for everyone to come home. Few people are aware that those expressions come from the German "alle, alle auch sind frei" or "alle,alle [r]ein komm' frei".

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.

To ask who's 'it' in a game, according to the dictionary, it's - cé atá air? - So to say - you're 'it' - would you say something like - Tá ortsa, tú ? And how would you say - you're out ? Amach leat has more the idea of telling someone to go out of/to leave the circle, for example, but I'm not sure that it would mean saying to them that they have been counted out. (I think I did ask this a long time ago at the other place, but it's no longer on-line).

Cé atá 'air'? lit. "Who is 'on it'?"
So: Tú/tusa atá 'air'. or Tá tú/tusa 'air'. "You're 'on it.'" - You're 'it'.

Go raibh maith agat - incidently, it might a silly question, but why is there an apostrophe at the beginning of - air ?

No apostroph, just quotation marks.

Ceart go leor.

And the card game called 'Old Maid', would that be - Seanchailín or Seanmhaighdean or perhaps there's a completely different name for it ? (in French it's le Pouilleux - the one who has fleas - or Mistigri and - you're 'it' is - 'chat' or 'c'est toi le chat')

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