Most people go the EB1C green card route after they are in the U.S. in L-1A status but that's not required; the entire green card process can be done while the applicant is outside the U.S. and not yet employed by the U.S. company.
Is it recommended or not? I've heard anecdotally that going through the Consular services to obtain EB1C is not recommended - because you can get denied easily without any protections you might get from being in the US.
You don't have protections in the US per se. If you are on a visa, your visa will remain valid and continue on its own terms. Your GC application is orthogonal.
