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Threshold bar and there are a lot of nicer options than the one you have pictured. You could get a larger nicer looking one that would hide the poorly trimmed edge. It's not rocket science to fit, just trim to length with a junior hacksaw.
You can pull the carpet out with a screwdriver and you'll probably see how it's fixed to the floor, push it back in with a screwdriver or pry bar preferably.
We’ve had new carpets fitted - which looks like it was done as quickly as possible. It wasn’t cheap but as well as this I can now feel the floorboards below in each bedroom, there are squeaks and floor doesn’t feel level. They also didn’t fit any of the doors back so we had to do that. Am I wrong to suggest that I don’t pay the 2nd half of the money until it is completed to a satisfactory outcome? Also the underlay the removed was thick kinda rubber material and think they’ve whacked in something much less dense - would that seem likely?