Will the background process exit when you log out in linux?

disown the job and it won't. If it's the only thing running in the background, you'd do disown %1. Otherwise it would be the job number given in square brackets when you start the job.

I'm not sure the exact answer to this question.

I do know that to detach from the console requires more than just redirecting to /dev/null. You need to consider using setsid as well as redirecting 1>/dev/null and 2>/dev/null.