You can be either on H-1B or L-1B to file a GC under either the EB-2 or EB-3 category, depending upon your education//experience and the position that your company has in mind. Either visas will do, but the H-1B allows you to remain in the US for an initial period for 6 years, while the L-1B is restricted to 5.
To start with, visit the US Consulate website/Department of State website. This is exhaustive and will give you a fair idea. Your employer's attorney should also provide you with some directions/guidelines along with the original Approval Notice/Notice of Action and a complete copy of the Petition that was filed. Please carry the latest pay stubs from the employer.
If your son is a USC then he can sponsor her after marriage. This would make it easier for your son's girlfriend to get a US-not difficult. You may have to file an affidavit of Support if she is to get Permanent Residence. This is required so that the govt. knows that she will be financially supported in the US and will not become a public charge.
If your petition A gets approved (I am assuming it was filed last year), you will be exempt from the Cap for petition B. In my opinion, once you have been counted against the quota, you are cap-exempt. This argument will have to be made in the cover letter for petition B.
In any case, company B will have to file a H-1B petition (if H-1B from A is approved). It does not matter whether you call it a new H-1B or a transfer. The only difference is that the transfer will allow you to start...