Rats, sorry you had troubles.
I tested and thought I had fixed picking up items before getting the quest has granted - I'll have to test that again but I thought it was working, As for the keys, due to some not-great setup you needed the keys and tools before you could use the crane
As for not being able to use it on the car, there is a delay before you can fix the car and you should get a message that the rubble is cleared then you can interact. Maybe I made it too long... though