Some employers will pay the cost to get a security clearance. Others will not. IME, employers will tell you in the job description which kind of employer they are.

"Must have a currently active $TYPE clearance." and "Must be eligible for a $TYPE clearance; position contingent on acquiring a $TYPE clearance." are the sorts of phrases to look for.

Depends on the company. My company prefers you to have one incoming, but since few people do, we sponsor most people.

I think many Aerospace jobs that aren't directly Defense/Military (think SpaceX Falcon 9 flight control development) are also behind clearance. Of course you will indirectly help launch military satellites but I wouldn't call it a military job exactly.