Is it normal to have swollen fingers after carpal tunnel release surgery?

For a while. Yes, postoperative swelling is to be expected. Elevate your arm/hand and keep salt intake to a minimum. Use a cold pack in the first couple days. Begin therapy or exercises as advised by your surgeon. Swelling should be minimal by the time you get stitches/staples out. If not, make sure you notify your surgeon.
Yes. Mild swelling is not at all unusual. However, contact your surgeon if the swelling is severe and/or painful.
No. Normally, after carpal tunnel surgery you will have a splint with instruction of elevation of the hand and exercise of fingers.If this instructions are not followed properly you may have some swellig of your fingers.