1. Can I leave my current university position but still legally stay in the once the company files H1B for me (no approval yet)?
as long as the new company started filing your application, you are fine.
2. Can I start my new position without my new H1B approval?
NO.
3. Is 14-day a guarantee to get H1B approved?
that really deponds upon LUCK. why in such a hurry?
Thank you very very much for your kind help.
all based upon my own knowledge, just for your reference, better contact with a lawyer for sure