They’re both extremely similar. Stay safe emphasizes the person is safe now and you want them to remain that way. The only situation where that fine distinction could matter is if you are talking to someone who is currently considering doing something potentially unsafe or is in the middle of it. Then you might say be safe instead. Again it’s a fine distinction and even my example is pushing it to find a difference.

Both are very much similar meanings. Both phrases are stating to remain safe; avoid danger. Do not harm one self, be careful. Both phrases are showing your concern and wellbeing to specified person whom you wish to be safe.

Nope.

Stay safe, or keep safe would both work.