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Yes

> if you don't make the source public

I'm not sure what you mean by that, the GPL only has constraints for redistribution. If you don't distribute anything (publicly or not), there is no way to break the GPL.

Assuming the code you modify is protectible in the first place, then yes, this sort of trivial modification or derivative would be covered by GPL and have to abide by its rules.

But just because you slap a license on something does mean you actually can enforce it.

No because you only need to make the source available on request from the original developer(s).

Yes. If you copy and modified then code, it doesn't change the fact that you copied it.