I do a fair amount of work with leather. You certainly can cut it up into placemat-sized rectangles. I do that type of cutting with a rotary cutter. You might want to round the corners- not essential, but I think it could look better. You should certainly burnish the edges.
You treat them with a burnishing agent such as the Tokonole that Hideout Hen suggested. That is one of the best ones. It works well & unlike a lot of others, has very little odor. You can find plenty of YouTube vids like
This on how to burnish. It isn't hard to do.
Other tools you may want include a corner-rounding punch, a mallet (to use with the punch), an edge beveler & a burnishing tool (sometimes called a slicker).