Best to do the first first and get to the second after this one is done.
i) Don't borrow trouble. If friction is not mentioned you can usually assume that it isn't there. Note also that the "line" between the car and trailer is ideal, that is to say that the tension on the car is the same as the tension on the trailer.
ii) The tension in the connector cancels out when considering the car and trailer as the same system. So do your Newton's 2nd by considering the system as a single mass.
iii) For there to be no tension in the connector then both car and trailer must have the same speed, non-zero in general. So if the trailer has a net force on it, what must the net force on the car be?
Can you finish it?