If I have had surgery for a bleeding ulcer in my stomach is it possible that it may start bleeding again?

Yes. Although not common, recurrence is a possibility. Be prudent and use antacids and follow a diet that will not promote ulceration. Also be sure that you have been tested for h pylori.
Yes. Depends on the correction and whether area removed, oversewn, bypassed, vagotomy, gastric, duodenal. Etc.