This is my selected bone which should deform only painted part of mesh. But when I move it, it also move some other parts - nails of other leg as seen on second picture. This is not only case, some other bones also move some other parts of mesh which shouldn't be deformed by that bone. I checked weight painting and those nails aren't painted with this selected bone (DEF_Front_Leg.R.003). Maybe worth noticing - I was mirroring vertex groups, but this was appearing before that also. What could possibly be problem and fix
First, colors are not a good way to judge weights. You're not going to be able to see the difference between 0,0,1/255 blue and black. And if weights are even less than 1/255....
Second, realize that more bones than DEF_Front_Leg.003 are transforming. Everything parented to (or somehow constrained by) Front Leg.003 are also transforming. If your dog's left front nails are weighted slightly to the right foot or shin bone, then those bones are also moving.
When you have problems like this that look like weight problems, you should inspect a few vertices for their actual weights. Stay in weight paint mode, but switch to vertex select, and select a vertex that is deforming inappropriately (like a stretched vertex on the left nails.) Then open a sidebar and look on item/vertex weights. This will list all vertex groups to which the actively selected vertex belongs, along with their corresponding weights. You can use this to figure out which is the wrong bone to which it's weighted; you can see the group, even if it's a very small weight value.