doorshingesmetalrust
There is a broken hinge which is stuck:
To compare, a good hinge is removed easily like this:
I tried to remove broken hinge with a hammer, however it is stuck very firmly. Is there any method to remove the broken hinge? The rust makes removal harder.
Take pictures and measurements. (Measure EVERYTHING on the hinge - height/diameter of pin in particular.)
You should be able to find the identical hinge with a bit of effort. Replace the part attached to the wood, and you should be able to drop the welded portion into it. Yes, you'll have to throw out the upper part of the new hinge, but it's the easiest fix and will restore the gate.
It looks like a lockring is used to hold the screw in place. You'll have to hold the screw still and rotate the lockring, or vise versa.
Using a flat-head screw driver and mallet, is a common way to rotate lockrings. Though, you'll have to find a way to hold the screw steady, while you bang the lockring free.
It's possible that the lockring is made of some type of softer metal, so grabbing it with pliers could easily damage it. Using a screwdriver and mallet applies force only in a useful direction, whereas pliers apply non-useful forces required for gripping. These non-useful forces may damage the lockring.
Best Answer
That style of hinge is generally made from a pair of metal cups and a piece of round bar.
It definitely looks like a good candidate for heat. There will be damage done to the surrounding paint. Either warm the top piece with a torch, or cut the whole thing off and weld in a new hinge. It looks as if there's welding to be done there anyway to re-attach the hinge to the door.
If a torch isn't available but a welder is, and if you'd really rather not cut off the lower piece of hinge, then you could weld a couple beads up the exterior of the top portion and grind them off later. As an added bonus such beads might give better grip to the locking pliers you'd want to use to grab and twist the hot hinge.