Will ovulation still occur alternately after a salpingectomy or will ovulation be more likely to happen only on the side with the remaining tube?

Still alternate: Even after removing one tube, ovulation will continue to occur from each ovary. It will not be more likely from the ovary that still has the tube. So a woman with one tube gone but both ovaries will usually take longer to conceive, since she only has a chance to conceive when ovulation occurs in the ovary that has a tube next to it.

Either side: A salpingectomy is the removal of one or both fallopian tubes, usually due to an infection or ectopic pregnancy. If one tube is removed there is no effect on chances of pregnancy since the other tube can catch eggs from both ovaries. If both tubes removed however, you'll need in vitro fertilization. There is no expected change with periods or chance of a healthy pregnancy once pregnant.