Menu block doesn't appears where I expect it I have this kind of menu:
"Level 1" has menu item "Level 2", which links to the page with "cat1" url. Then, I have node which have aliases "cat1/[nid]", hence, in my opinion, menu block (provided by "menu block" module) should appear. But it doesn't. Why?

If you haven't worked with menu block module and not familliar with it, then how can I force to show "Level 1 -> Level 2" menu part if a node with alias "cat1/[nid]" is loaded?
Solved that! Inspecting the code of Menu Block cleared it out for me that this module doesn't work with URLs. What a pity!

But: 

There is a completely awesome bunch of modules: Context withContext:Menu Block which make it possible.
Context menu block modules make Menu Block module being aware of context. Also this page may be useful.

But still, if you have better solutions, please write it here.