I think we covered all these issues above.
1. See Drucker v Mauro and Riverside v. Monroe.
2. She was asking for you to be on the lease pursuant to succession rights, not as an additional tenant. As a family member you will have succession rights when and if it becomes necessary (i.e., when your mother passes or moves). Until then it's premature. Still, as a son you don't fit as an additional tenants that would trigger a new vacancy lease. As for your proof of residency, again, this is premature. You do not need to prove your residency at this point as you haven't asserted succession rights and it is not at issue. It will not be an issue until your mother passes or leaves.
3. SCRIE has nothing to do with your lease issues. NOTHING. All household income must be reported and that will determine the SCRIE subsidy to the rent. See the SCRIE rules for that. But it has nothing to do with your lease.
Make sure that the people helping you with the response to DHCR know what they are doing. If they aren't experienced tenant attorneys or activists, I'd be concerned.
Go see the lawyer this week, not next.