Is it normal for my knee to get really stiff after a total knee replacement?

Yes. The pain associated with total knees makes most patients struggle with bending after surgery. This, along with the healing process of the large incision, causes significant scaring, reducing the patients early motion. This requires fairly aggressive and regular exercise and stretching to lengthen the tissues while they are healing so that you can regain your preop motion which is the goal.
Yes. It should bend more easily as it heals. If it is not improving, i would see your surgeon.
No. Range of motion after knee replacement should be similar to what it was before surgery. Loss of full range of motion can be a sign of arthrofibrosis, caused by excessive scar formation. The cause is unknown. Possible treatments include knee manipulation, arthroscopic debridement, and/or revision of the tibia polyethylene insert to a thinner insert.