Yes. The UTI probably caused some pain while in the litter box so now he is hesitant to go in it. Cats don't realize that it's the UTI and not the box that caused it. This will improve with time and patience on your part.
And, yes, I would recommend shaving him for the summer if this is an issue. Hair can cause an intestinal obstruction, particularly if he is losing as much as you say he is.
You might also want to talk with the vet about using an antihistamine to control some of the hair loss if you suspect that it's caused by allergies.
Let me know if you still have questions.
