Im certainly no vet so take my advice with a grain of salt, but it strikes me as completely unreasonable how your wife controls and limits access to your sons. I understand that your past anger caused her to be afraid for her and possibly your son's safety, but she is obviously no longer really concerned about their safety if she lets you take them back home without her for an extended visit. I think it is obviously a power play by her and your sons are the pawns. I recall that she did this before where she only let you see them for a very short period for no good reason. I would not stand for this. She gave in about Saturday only because the boys are with you and there is not much she can do about it, but I bet that she will seek revenge by trying to limit access to them in the future. If she tries to instigate a fight about the extra day, or later this issue comes up again I think you need to tell her, calmly, that you want (as is your right) to see your sons more, that they need to see their father more, and that it is unacceptable for her to hold this over you. She really does not have any real power here - you can force visitation through legal means if necessary. The only power she has over this issue is what you give her. Her power is that she can decide if SHE will see or speak to you. You should make sure she knows that you will respect her wishes on what she wants to do with her own life, but it really isnt for her to say whether you can or cant see your own sons.
That said, I think you should do all you can to help her out with the car. If she is driving around with your sons in a car that wont start or stalls, that is unacceptable and unsafe. Despite everything, you are still her husband and their father. You love them and ensuring their safety is the most important responsibility you have. Either switch cars this weekend or get her another car. And while you can offer to help her go car shopping or something, dont let her think that you are looking for some sort of attention reward for this - you are doing this because she deserves a decent car and you want them to be safe.
As for the respect thing, I dont really know if that is applicable for you. It seems that your wife left because she feared you, not because she thought you were an unattractive, conflict avoidant, spineless beta male. Stand up for your rights, set reasonable boundaries concerning things that directly affect you (finances, child contact etc.), but I would try to do so in a gentle, loving manner and always explain that your motivation is one of care, not control.
