"I cannot sell my car."
Translation:Nie mogę sprzedać mojego samochodu.
I guess usually when hearing "Nie mogę sprzedać samochodu" you would suspect that someone is selling his own car, but the context could imply also other things, e.g. "Nie mogę sprzedać samochodu mojego brata" = "I cannot sell my brother's car" or "Nie mogę sprzedać skradzionego samochodu" = "I cannot sell the stolen car".
I think you shouldn't.
It seems that some misunderstanding has spread within this course. The possessive pronoun referring to the subject is not generally implied in Polish. If it was, there would be no need for a word like "swój" (with all its forms). It is often implied while speaking about body parts, family members or people like "szef", "sąsiad" etc. But I think with most nouns you should use a possessive pronoun if the context doesn't show who is the owner.
I wouldn't be so sure... it's a good argument about the need for "swój", but still, I'd rather say "Próbuję sprzedać samochód" or "Zaprowadziłem do naprawy rower" (I took my bike to the repair place) than use "swój". We may say that that would translate into English using "the car" or "the bike", but the sentences are still actually about "my car" and "my bike".
The difference between this and the body parts is that the possessive with body parts is not even redundant, but simply weird. This sentence is also not as obvious as those about family members. But I still think we can safely accept omitting the possessive here. Added.