Yes, they are.
Наша группа останется (perfective - future tense) в Тбилиси до вторника. - I would use this one to just convey the fact that the group stays in Tbilisy till Tuesday.
Наша группа остаётся (imperfective - present tense) в Тбилиси до вторника - I'd use this one to do the same thing as written above but this phrase also implies that this stay till Tuesday is planned.
But those two are kind of grey area, because they can be used interchangably I suppose. Depends on a particular person I guess
Маша пригласила меня на дачу, но я не мог поехать. Пригласила is used because you just imply the fact of being invited, that doesn't matter if you accepted it or not.
Не мог поехать is used because не мог ехать sounds like you didn't have the ability (not possibility!) to ехать at that moment
One more thing, I think your Russian is already good enough to tell you this. I hope it won't confuse you. If it does, just ignore the following:
Маша приглашала меня на дачу, но я не мог поехать - this one is also possible but it carries a little bit different meaning. Here it sounds like Masha was inviting you to the dacha very persistenly, in other words she did it more than one time.
I imagine it like that:
Masha: Let's go to the dacha, shall we?
You: I'd love to but I can't
Masha: Maybe you can after all?
You: No, I can't
Masha: Please, let's go to the dacha...
I hope this helps