I have two concerns:
1) Is the tax code of H correct even though the rollover was to another Roth 401k, not a Roth IRA?
2) The 1099-R lists me as the recipient, even though the checks were made payable to the firm managing my 401k plan. Is that going to cause any issues that may make it look as though I took a distribution?
As you found, code H is for a direct rollover from a Roth 401(k) to a Roth IRA and is incorrect for a rollover from a Roth 401(k) to another Roth 401(k). The correct code for a direct rollover from a Roth 401(k) to another Roth 401(k) is code G. You should contact the old 401(k) plan administrator to obtain a corrected code H Form 1099-R indicating $0 distributed as a Roth 401(k) to Roth IRA direct rollover and a new original code G Form 1099-R reporting the direct rollover of the Roth 401(k) to another Roth 401(k). The rest of the reporting on the code G Form 1099-R should be the same as on the erroneous code H Form 1099-R, so there is no tax difference between the two.
As it stands with it reported with code H, the IRS will likely find a tax deficiency when they do not eventually receive a Form 5498 from a Roth IRA reporting receipt of the rollover contribution and they instead treat the distribution from the old Roth 401(k) as a taxable distribution (until you provide documentation substantiating the rollover to the Roth 401(k)).
It is correct for you to be listed as the recipient because you were the owner of the account..