Is jaw alignment for underbite painful?

No. Orthodontics and/or orthognathic surgery should not be painful.
Recovery required. If you have orthognathic (jaw) surgery to fix your underbite you will need to be prepared for a healing time of at least one week. Make an appt with an orthodontist who will refer you to and oral surgeon if its seen that you can benefit from the procedure.
No. Depends on how it is done. Most patients relate that 3rd molar (wisdom teeth) extraction is much more painful and uncomfortable than corrective jaw surgery. Also, most patients comment is that they wish that they had done it sooner. The younger you are, the quicker you heal and feel better. You should always consult your orthodontist and an oral surgeon that does this type of surgery all the time.
Not too bad! My patients tell me that orthognathic surgery results in only moderate discomfort and is predictably managed with contemporary pain management techniques and drugs. Of course, it depends on what exact operation you're discussing....For an overbite, that could be moving the upper jaw forward, or moving the lower jaw backward, or both, which of course would result in more pain but still manageable.
For a short time. During surgical correction of an "underbite", the patient is anesthetized (put to sleep) so you wont feel much. For the next few days, you might be sore.
No. The better your Orthodontic Specialist prepares your dentition, the less the Oral Surgeon will have to do. Jaw surgery is done painlessly under general anesthesia. Medications will control soreness during the healing process. Start with the best Orthodontist you can find, who will pick the best Surgeon to resolve your problem.