What should I use to fill big gap around door frame?
With the deadbolt on the door, it appears to be an exterior door, so the space in the jamb should be insulated, both at the top and the sides. No need for a full roll of insulation to be bought, there are smaller pieces that can be bought such as fiberglas pipe wrap, or ask a person you know that may have a little on hand. It won't take much. 

The drawing below will explain a bit on what to do about sizing material for the trim to cover the gap. Perhaps other trim that is throughout the house will give an idea how to treat the door.


Perhaps the trim in your place looks like this? The sides of the door in this pic looks like the trim will fill the footprint in your flooring.

Carpet trim is what I Googled to find the strip to cover the edge where the flooring is cut at the door.