~하지 말 걸 그랬다 or ~하지 말았어야 했는데 is can be transferred to "I Should not do that in the past" ~할 걸 그랬다 or ~았어야 했는데 is "It would be better if I do that in the past" Yes they are regretting, however, one is you did something, other is didn't do in the past.

Ah, I misread the two first as one. Then the second is the negation of the first. And the THIRD is the same as the SECOND in my previous comment.

And the third?

I'm not entirely sure, but I think the first one is about regret for something you didn't do, second (edit) for something you did do and the third (edit) one is about something you had to do (whether or not you did it). But I may be wrong.