Is it possible to make special commands like cleaning cache of Symfony on remote host automatically after upload files by Ctrl + S?
Currently it's not possible.
Watch these and related tickets (star/vote/comment) to get notified on any progress:
Right now I may only suggest creating a custom Remote SSH External Tool entry which you can invoke when upload is complete.
NOTE: You can assign a shortcut to such External Tool entry which you can press instead of invoking it via menu/toolbar button etc.