Most time company will apply H1B after you say "yes"

Not before, There is a time to start new hire procedure. Like check your background, etc. Also If you have a H1B, company will do a H1B tranfer. You can start new job as soon as it filled. If you don't have one, you need wait it approved(maybe wrong). If there is no space (start each Oct, ??K total/year), you have to wait until next Oct.
You can working on 1+ H1B. But most company want you only working for them, otherwise will fire you.