H1B Transfer help

[kkp2208]

Dear All,

in the year 2016 I was working on L1 Visa with one of the giant consulting company from India,

one employer A(small consulting firm) filed H1b Visa(L1 to H1 CoS), The I-797 document having the Employer name A,

I started working for employer A from Dec 2016, now Employer A generating my pay slips with a different company name B, when I asked, my employer saying that the organization A merged / acquired and rebranding with B, so the payslips are running with new company B, and filing H1B amendment...etc.

Meanwhile I got an offer from a Non profit employer C(reputed us company), who is willing to do H1B transfer not premium.

need to submit documents to Employer C,

Need help on below questions:

I-797 is having employer name A, and payslips has employer name B, in this situation will there be any issue with H1B transfer with employer C?,

What kind of issues or additional documents uscis may ask, getting RFE or reject...etc regarding I797 employer name vs Payslip name mismatch?

employer C asked to join as soon as they receive the receipt number, since now H1B transfer process is taking more than 6months, by the time uscis responds  I will be working for employer C for around 6 months, is there any chance no questions on previous company and max chances are getting approval since it is reputed firm. pls share your thoughts?

if I need to travel out of country can I travel during H1b transfer in progress?