I know the Roman Empire briefly conquered and occupied what's now England, but did they ever reach Scotland? Was Hadrian's Wall really built to guard Roman Britain from the Scots trying to take their land back, like I've always thought?

Yes the Romans did go into the highlands of Scotland even fighting a fairly large battle near modern day Inverness. Also before Hadrians wall was built there was a wall to the north called the Antonine wall which was made from dirt and wood. Hadrians wall was never really meant to keep the celts out indefinitely as they could easily get past it or through it in a day. Instead it was more a symbol of Roman power, skill etc to mark their domain. Also remember that there was quite a bit of trade between the Romans and those north of the wall hunting dogs were a big commodity bought by the Romans and wine and luxury goods were traded to the celts

Edit: Good proof of it being mostly for show is around mile castle 37 where there is the full height wall on the top of a huge cliff, Hadrian more or less designed one part of the wall and it was copied and pasted all the way along