You must have multiple trackpoints, or multiple tracks overlapped. The select point tool selects a single point and gives you the option of moving it, deleting it or inserting points on either side of it.
Try removing all of the files except the one you are working with to get rid of duplicates. It may be that you are downloading incorrectly from the GPS -- if you "save" a track you can get multiple copies of data.
The profile tool has a black arrow and with purple waves (a little profile, sort of). It's next to the select point tool (arrow).