The first and third do not work. 'For' in this sense needs to be followed by a noun/pronoun, as in your second sentence.
Incidentally, your sentence would sound very odd in BrE. In Br E, public schools are very expensive private schools.
Can one use:
1-He is good at French for going to a public school.
if the intended meaning is
2-He is good at French for someone who has gone to a public school.
In general, could one use "for + gerund" instead of "for someone who" (given that fact that.
I guess this could be another example:
3-He speaks English well for growing up in a non-English speaking country.
(Meaning: He speaks English well for a non-native speaker.)
I do not think 1 and 3 work. In those sentences 'for' seems to give the reason for something.
You could use 'for' but not with that syntax: 'He speaks French well for someone who went to a state school'.