out of status
2. Should I ask the company to apply for H1b transfer, or petition a new H1?
Transfer H1b = new petition. You can have 180 days gap between the two h1b. However, you may need to go back to home country to get your new h1b. Fortunately, you are qualified for that because you can have total 180 days during the total six years h1b without leaving US.
3. Can I start to work after I the petition, or I have to wait for its approval.
It is better to get approval.