Not an easy question. The legal status of a region usually depends on what functions the local goverment is implementing. It affects what taxes the region can claim for itself and what it should collect for the federation, what actions can or cannot the local goverment do and so on. Usually such status is determined through some kind of negotiation and depends on the economical and cultural properties of the region. Yet, the conditions mentioned above can be different for particular regions even though they are named the same way as others.
As for politics, any region or even a city can have it's own laws in particular spheres as long as they don't contradict the laws of the higher level.

Yes, as a citizen of the RF, you're allowed to live, work and travel anywhere across Russia. Yet, you should keep in mind that cultural traits of some regions may differ significantly.