Once a hair transplant is done can you still get crew cuts?

Yes. Your best option is to have fue hair transplantation. In this procedure the hair is removed one by one from the back of the scalp so that a scar is not left behind.
Depends on technique. It depends on the technique used to harvest and implant the hair grafts. Strips or plugs are easily visible and crew cuts should be avoided. Current methods of fue are improving both the donor and recipient sites and crew cuts are possible but it may take at least 6 months to assess the outcome of the procedure and early manipulation of the grafts should be avoided.
Yes, if done well. If the hair transplant is done well with micro-grafts in the recipient it will look natural no matter how long your hair is, also important is the appearance of the donor site scar. So, check out yor surgeon and his/her results carefully before you commit.
Yes but. You will want to make sure you discuss the details of the transplant with your surgeon. If you have follicular unit (fu) grafting the resulting appearance will be very natural, even with short hair. Most surgeons use fu grafting up front, but often transition to larger grafts further back in the hairline. If your hair is very short, the large grafts may be more noticeable.