Doors – What are these fasteners on this hinge

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.

1. Remove trim.
2. Remove door casing.
3. Remove drywall around jack studs (2x4 closest to door opening on each side) and pretty much all the way up to the ceiling. Stop if you have roofing structure above your door - keep going if you don't.
4. Pull out jack studs (you could just extend them with a piece of 2x4 on non load bearing walls)
5. Push up header to desired height. You will have to take out the cripples (small 2x4s) above the header. I don't think the door in your picture is going up a foot with that trim.
6. Put in new longer jack studs on each side.
7. Install new, longer door.
8. Drywall, mud, tape, all that good stuff - this might take the longest.
9. Install new, longer trim. You could use old trim if you could find some casing that will fit on the bottom corners and if door is same width.