If you are working in the U.S. with an H-1B visa, you have up to 60 days or until your I-94 expires (whichever comes first) to find a new sponsor for a change of employer, apply for a change of status, or leave the U.S.
Extensions are possible if an employer files a PERM case or an I-140 petition on your behalf at least one year before you reach the six-year limit in H-1B status. Thereafter, your H-1B status may be extended in one-year increments. Once you have an approved I-140 petition, your H-1B status may be extended in three-year increments until your priority date is current.
The new employer must file an H-1B petition on your behalf. You may legally work for the new employer upon filing of the H-1B petition.
No, a U.S. employer must file an H-1B petition on your behalf.
A "specialty occupation" qualifying for an H-1B visa is any job that requires the “theoretical and practical application of a body of highly specialized knowledge to fully perform the occupation AND requires the attainment of a bachelor’s degree or higher in a specific specialty as a minimum for entry into the occupation in the United States.” The position must require a bachelor’s degree or higher in a specific field and the beneficiary must have the required degree.