Is it possible to change board side for components group without breaking their relative positions and existing interconnections inside this group?

No, it is not possible at the moment.

Please consider this feature for the next versions, because it helps to do bottom-up layout design.

Also i found, that it is difficult to move already routed components group relative to another one, because existing routing breaks in a strange way. It will be very helpful to have something like "local lock", which freezes relative positions of traces and patterns inside the group, but allows to move such groups relative to each other.

Ok, we will consider this. We plan to add option to save complex patterns (few patterns with vias, routing, etc.) to library in future versions. Moving group of components from one to opposite side of board is similar to moving of complex pattern.

Heh, I know a local PCB manufacturer that has made their own Gerber editor just so they can add exactly those sorts of quick fixes upon panelling.

I needed to make a stencil for a 2 sided board, and wanted everything on the same side of the stencil. To accomplish this I ungrouped all the patterns, then grouped the pads from all parts together into a single pattern. I could then change the side of the board without problems. This would probably cause issues in a design that needs to be updated by the schematic, but it worked well for this purpose.