Can you still get pregnant after a partial salpingectomy?

If not bilateral,yes. Mother nature was "great" on redundency by providing two fallopian tubes and two ovaries for conception. If one is not available the other will "do the job".
Possible. 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.