See if any of the hostels will accept infants in a private room. There are quite a few hostels and some have private rooms with bathrooms in suite or shared with only one other room. Almost all the hostels are in good locations or near relatively centrally located metro stations.
Backpacking in Russia with an infant is probably no different from traveling elsewhere with a child that age. One exception, I'm not sure how useful a stroller would be because there are lots of under passes which are not ramped. OTOH it depends on how much your 7 month old weighs as a stroller can be useful as a portable place for napping.
Don't expect anyone to have car seats. Some places won't let you enter with young children.
Try searching various blogs about parents with young children in Moscow for ideas of things to do and parks to visit.
Ruth