For the first one
A, the first and third ones are correct
Tom would go to high school until last year.
^ is technically correct but the sentence does not flow well in a way
It is more grammatically correct in one writes
Tom would go to school last year.
I doubt "Tom was going to high school until last year." is a correct sentence. Is my understanding correct?
^ This sentence might be correct, I think its grammatically correct, but it sounds weird, so its better to stick with the other ones you have mentioned.
B, all 4 correct, as they all imply how Tom dated Becky using past tense.
For the comment on the following:
But I still do not understand why "would go to school until last year" is weird while "would go to school last year" is OK. Would you explain what makes this difference, please?
To be honest, both these sentences actually have different meanings, "would go to school until last year", implies that Tom stopped going to school at the start of last year, and "would go to school last year", means Tom used to go to school last year and ended school probably at the end of the year.
E.g. The current year is 2022, put the first sentence into perspective (would go to school until last year), then it means Tom stopped going to school at the start of 2021. To clarify, "until" means "up to", so Tom stopped going to school when 2021 started. The second sentence (would go to school last year), implies that Tom stopped going to school when 2022 started.