Is it important to remove a splinter asap or will it just come out on its own over time?

Splinters: Usually remove it a soon as possible, if it stays in too long the skin grows over it and it is harder to remove ( only rarely does it get extruded).Depending where it is can cause pain or get infected ( especially DM is complicating the picture)

Got tweezers?: It's a lot easier to get a splinter out if you can grab it right away. Tweezers are the best bet. Otherwise it is likely to break off, soften and become embedded in the skin. The skin will eventually slough off, and eliminate the splinter, but this can take weeks.

Splinter: depends where and how deep the splinter is. some splinter from the bullets may not need to be removed because hey are embedded deep and could be sterile for all practical purposes. other like wood and other contaminated superficial splinter need to come out as they will cause infection.