Yes they can. If they’re sick too or anyone has had a COVID test done then they should stay home but otherwise send them.

Yes of course, providing it’s not something highly contagious like covid 19 and your kid has not been tested for it. Also if someone else can care for your kid you go to work, although if my kids are really miserable I don’t leave them with a sitter, not fair on the kid or the sitter.
Most kids will try skip school if a sibling is staying home sick, mine sure do.