To be honest, I'm very novice in Emacs, so is it possible somehow to bind TAB key in insert state of evil mode to shift text after the cursor right on current indentation amount of spaces at each key press. And how to do reverse process (shifting of text after cursor to the left) on Shift+TAB pressing.
In vim, one would use Ctrl+d and Ctrl+t in insert mode for this, which is replicated in Evil.
That being said, you would have to rebind functions
evil-shift-right-line to TAB and Shift+TAB.