Can I get compensation if a trailer rental company’s mistake caused damage to my car?

You could sue U-Haul for negligence.  Negligence is the failure to exercise due care (that degree of care that in this case a reasonable trailer provider would have exercised under the same or similar circumstances to prevent foreseeable harm).  In order to prove negligence, you will need to prove duty of care (discussed above), breach of duty (failure to exercise due care by providing the wrong size trailer), actual cause, proximate cause and damages.  Actual cause means but for U-Haul providing the wrong size trailer, would your car have been damaged?  If the answer is no, which appears to be the case, you have established actual cause.  Proximate cause means were there any unforeseeable intervening events which would relieve U-Haul of liability?  If the answer is no, proximate cause has been established.  Damages means the amount of compensation you are seeking in your lawsuit for negligence.  Your damages would be the cost of repairs to your car and the cost of commuting to school and work.  You will need to mitigate (minimize) damages which means that the repair shop you select for repair of your car should be one whose charges are comparable to other repair shops in the area.  If you were to select the most expensive repair shop you could find, your damages would be reduced accordingly.  The same applies to the commuting cost you mentioned.  If there is a reasonable, less costly alternative, you would have to use that or your damages will be reduced accordingly.

Prior to filing your lawsuit against U-Haul, it may be possible to settle your case with their insurance carrier.  If you are dissatisfied with settlement offers from their insurance carrier, file your lawsuit for negligence.  If the case is not settled, you will need to file your lawsuit prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.