Hey Dmitry not sure what the problem is exactly but for starters, you have the checkbox “use custom template”, but then you don’t have any filename specified…
Try it with search.php
and see if you get some valid results? (I’m sure the layout will be wrong).
Regardless of this, I’m pretty sure you should be seeing the URL change.
In some cases when the URL does not change its because WordPress cannot redirect after submitting the form – and this is normally caused by a hidden error in WP, with your theme or your plugins…
Try installing debug bar and submitting the form, and try to see if there are any errors. If you have any errors, then these will need to be fixed first.
Let me know the results.
(Ps seems we are on different time zones, I won’t be back online now until tomorrow).