When the LL fails to renew your lease (either make an offer or return the executed lease), your rent does not go up. See the RTP-8 instructions.
FYI, my LL has refused to give me a lease for several years, and my rent is the same.
And if the LL refuses to cash your rent checks, just make sure you send them by either Cert. of Mailing (cheaper) or Cert. Mail. That proves you tendered the rent. Does he just hold onto the check or return it? Either way, make sure you have the money in the bank and DO NOT spend it. The LL is hoping you won't have it later.
As for the new documents, to the unknowing eye, yes, they may appear to be something that could be seen as reducing your rights. And yes, there are judges and DHCR personnel who buy into that notion.
Beyond the fact that you they might be seen as reducing your rights, you DO NOT have to sign and/or return such documents, except in a few cases, which we have covered in this and other threads.
The law is quite clear. Even if a private agreement with the LL appears to expand your rights, it's still illegal under the RSC. Renewals MUST be on the SAME terms and conditions as the expiring lease. See Riverside v. Monroe at: viewtopic.php?t=7883