"I have tried and finished my homework" is correct to my ear (possibly because I'm Canadian), but it means successful completion as opposed to "I have tried to finish my homework" implying I didn't get around to it.

That's not what "try and" means though. It's perfectly fine to say "I'm gonna try and fix this" when you don't know if you can fix it.

(Source: I say that shit all the time).

I would only say "try and" if I thought it was likely that I'd at least make some progress towards the goal.

If I expected failure, I'd instead say "try to" fix it.

Maybe, but even if true, it's still very clearly different from what the parent said.

To me, "I"m gonna try and fix it before I buy a new one, but that's probably what I'm gonna have to do" is a fine sentence.

For me, if someone says "try and see for yourself", it implies possibility of failure or something new.

If outcome is considered in terms of success or failure then try implies non-zero probability of failure. If outcome isn't considered in terms of success or failure then "try this flavour of ice-cream" is experience and try this outfit(fits or doesn't) is a test you can't fail. Philosophically, it is as master Yoda said: Do or do not, there is no try.

Then there's people like me who use the popular contraction for the first three words: "Ima try and fix this."

A contraction containing another contraction, in fact.

That sounds horrible to me. I’m going to try to fix it.