Other options for nothing are faic, tada or dada used with the negative. e.g. Níl faic agam = I have nothing.
Well, kinda. As you can read in the link provided by scilling, "chaque" is ultimately a crossing of two words: "quisque unus" and "catunum" < "cata unum".
"gach" appears to be related to Latin "quisque" (http://www.ceantar.org/Dicts/MB2/mb19.html, so the original connection is rather from Proto-Italo-Celtic), so, in a way, to "chaque", but there have been so many alterations and conflations and stuff that, to me, it's a bit complicated to say they're connected.
Irish doesn't have a verb for "have". Instead it uses the verb bí (tá in the present tense) and the preposition ag, with the object in English becoming the subject of the verb in Irish, and the subject in English becoming the object of the preposition in Irish. When that's a pronoun, it combines with the preposition to create a prepositional pronoun (agam).
This is outlined in the Tips & Notes for the 3rd skill on the Irish tree, Phrases.