I know that a piano can accompany an orchastra (piano concertos). But is anyone actually in an orchastra playing the piano, or do orchastras just "hire" one?

It can happen like you say, the piano accompanies the orchestra but with piano concertos, the orch accompanies.

Occasionally the piano is just another instrument equal to the rest.

Orchestras play in halls where there's almost always a piano. Not always a good one. Major concert halls have a good piano.

The major professional orchestras will have a pianist, who will also be required to play other keyboard instruments such as the celeste.

I would think they will get paid per performance rather than a salary

I play tuned percussion in our university orchestra, and piano falls into that section on occasions. Sometimes it's just an instrument on its own.