I moved to Russia in 2022 and worked with 2 brokers (https://moscow-rentals.ru/ and https://moscowestates.com/).
I ended up going with https://moscow-rentals.ru/ as they were more professional. They are amazing. Their main contact person Ksenia speaks English, and they work on a commission basis. Once you sign the lease, you have to pay them the commission which is equal to 1 month rent. What is also important is to find a landlord who is willing to register you. Whenever you come back to Russia, you need to register within 7 days. Only your landlord can do it. That is VERY important.
We didn't go with https://moscowestates.com as their solution for registration seemed to violate Russian law, and we glad we didn't. They offer a "registration service", where they have someone who registers a bunch of people in their apartment (not where you actually live). That seemed shady, and we didn't do it. I am glad, we didn't. About 2 months after we moved, we got a visit by the police to check our lease agreement and passports.
As for types of apartments, be mindful that there are 2 types (kvatira/квартира and apartments). And apparently one can only get registered at a kvatira, not an apartment. Which is a pity as some of the newer building are apartments.