If I have obstructive sleep apnea, will having a breathing tube during open-heart surgery give me complications?

No. The fact that you know you have apnea... And presumably it is being treated actually decreases your risk of post-op complications. Untreated sleep apnea could result in post-obstructive pulmonary edema when the breathing tube is inserted to help you breath.
No. No. The endotracheal tube ("breathing tube") will bypass any areas of obstruction.
Not likely. Be sure the anesthesiologist knows you have osa. He will need the info when you are in recovery. A breathing tube can replace a c-pap if you are using one.
Not at all. As long as your anesthesiologists know about your apnea you should not have a problem.