Great, probably videos I saw on YouTube was in right direction
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So, if I have 2 layer PCB:
1) 1 layer only for Ground plane, no other nets should be located there?
Ideally, yes, but you may be forced to route some tracks there, if you can't route them all on the top layer. As a general rule, splits in ground plane should be avoided, so you can, for example, try to route them around the outer perimeter of the board.
For some circuits it wouldn't matter at all, for some it will. The faster the signals, the more important it becomes to have a good ground plane and proper routing.
There's a lot of good educational material on youtube with regards to ground plane and respective do's and dont's, you may want to look that up.
2) I saw that usually authors of these videos use F.Cu for GND Filled Zone, is there any common standard for this (I thought B.Cu is more suitable)
It will introduce additional parasitic capacitance, so watch out for that, if it matters. On the other hand, as far as I understand, it may be used to improve shielding, reduce crosstalk etc. It's a very deep topic.