What are the correct phrases?
1. "couple years" or "couple of years"?
2. "half year" or "half of year"?
It's a second language for me, but as such, I had to learn it more formally than more or most.
I agree with Gendji, that "couple of years" and "half a year" are appropriate in your choice sets, although I'd expand on karbok and suggest that both "couple" uses are informal or even slangy. Particularly in writing, I would use "two (or 2) years" or "six (or 6) months". If the intent or point is to be imprecise, it is better to say or write "about two (or 2) years" or "about six (or 6) months."
By the by, "half a year", in common usage or parlance, is quite rare: I don't believe I'd ever heard nor read that time frame applied. To wit, to me, "half a year" is slangy or informal enough that I'd barely distinguish it from "half year", although the context
could matter.