Or in other words, are the autonomous regions basically neglected when it comes to federal funding for schools, public transportation, utilities, etc. ?
Probably the opposite is true. The regions are not even, but it has nothing to do with its status. There is no intelligible system, but poor region will get more money if its chief has more 'weight' within central authorities. Moscow city solves many of the problems on its own because the big share of all taxes, collected over the country, flows into Moscow budget.

are you allowed
Yes, of course.