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 depends on whether the wood is actually singed or just the finish.
You need to remove the finish on the door (and perhaps the whole cabinet unit), but begin with a test stripping of just the damaged area. This can be done with solvent strippers (harsh and smelly, but faster) or citrus based (a bit slower but more person/environment friendly). The wood then needs to be lightly sanded.
Refinish
If the wood is not charred (black or severly discolored), you can strip the rest of the door, sand it, stain it to match the rest of the cabintet and finish it with a polyurethane finish with a sheen similar to the cabinet. Getting it all to match can be hard, so you may wish to do the entire cabinet.
Repair and Refinish
I the wood has a slight char or discolorization, you can try to sand these areas out or use a diluted wood bleach to match the rest of the wood. If this works, then proceed as with refinishing above.
Paint
If the damage is too deep, you need to cover it with an opaque finish, such as paint or a heavily pigmented stain. This will change the overall look so it needs to be done to the entire cabinet. Sand away the damaged finish and lightly sand the rest of the piece with 120 grade sandpaper. Then refinish according to the paint/stain directions.
Countertop
If the laminate is blistered, there is little that can be done except replace it. If it has just lifted, you may be able to reglue it. This generally will not work unless the lifted section is near an edge and can easily be pushed flat. If you can lift the whole section at least 1/2 inch, contact cement works best. Spread glue on both faces to be joined and let dry separately before pressing together. If the gap is too small, you can force conventional glues under the area and weigh it down while the glue sets.
Best Answer
When refinishing old surfaces like that, I have had great results finding the right compound to do what is called I think, re-amalgamation. It is when the proper solvent or compound I called it earlier, is used to melt the finish on the surface and allowed to dry again. It will remove some of the finish, or perhaps even most of it if too much is applied and wiped back off, but it requires no sanding, but you will need a respirator for the fumes. This saves the patina of the surface and will remove all the glass rings and minor scratches that was on the surface. If there are deep scratches it will not remove them, but it may color them in a bit so they are not so noticeable.
The finishes used typically are shellac and lacquer. Shellac can be removed with denatured alcohol I think, lacquer is cut with lacquer thinner.
If you really want it all clean and pristine and brand new looking, the sanding is the best way to go since the only way to get rid of deep scratches is by sanding, unless you want to try your hand at using lacquer sticks. Those are a heat melt-able filler that come in numerous colors to blend in with the color of the wood.