Is it normal for the doctor to push the fat out of the injection site with his hands?

There a variety of techniques used for fat transfer, and without observing the procedure itself, it would be hard to comment and give accurate information.  If the fat survives from 2-6 months, then you have a good chance of keeping the fat.

One day after surgery if very early to start worrying about the results.  Also, without knowing all the specifics of your surgery and seeing your pre-op photographs, it really is impossible to give you specific advice.  Typically with liposuction you can expect a moderate about of bruising and swelling which resolves over a few weeks.  During the healing process you may feel slight tingling or itching which is due to the skin healing.  If you are concerned about your recovery, I suggest that you talk with your surgeon.  Hopefully you were treated by a board certified plastic surgeon.  If not, I would suggest that you see the care of a board certified plastic surgeon in the future.  Good luck.

A burning or tingling sensation after tumescent liposuction is not uncommon.  In fact, most patients experience some degree of altered sensation in the area that was liposuctioned for several days or weeks after the procedure.  Regarding the technique used for your liposuction and fat grafting, it is unclear if this was standard procedure or not.  While traditional liposuction and fat grafting (i.e. fat transfer) involves the actual removal of fat from the body with cannulas, collecting and processing the fat outside of the body, and reinjection into the areas of concern, there are limited instances when surgeons may perform "liposhifting" to push fat from an area where traditional liposuction was performed to an immediately adjacent area, and vice versa.  Regardless of procedure performed, you will not see the final results for several months.    

Most importanty, however, is to ensure that your surgery, and any future plastic surgery or revisions, is performed by a plastic surgeon that is certified by the American Board of Plastic Surgery and a member of the American Society of Plastic Surgeons.  You can go to each of these organizations' websites to find a list of legitimate members.