Planking: Your wood veneer is old. The adhesive used to bond the top layer of veneer is old. It needs to be glued/bonded on again.
This is not the cheapest way to fix planking but I wanted wavy, planked cabinet sides locked down tight. Because it was a cabinet with both sides planked I had to get serious. I bought a large 8 oz. bottle of super glue. I laid the cab on its side and flooded the side with light viscosity fast setting superglue. The super glue flows under the planked areas and bonds everything super tight and it will never plank again. Then I did a light sanding, applied any Bondo to smooth things out, sanded again and then painted.
EDIT: I should add that I used one 8 oz. bottle of superglue per each cabinet side. At $27.00 a bottle, the fix was $54.00. But it stays fixed and is cheaper than a new cab.
CAUTION: If you do the super glue flood method, either make very sure you have enough fresh air or wear a mask. Super glue in quantity will knock you on your butt.
Broken cab corners: You have gotten many good suggestions.
I prefer to get out my wood chisels and cut the broken wood away, then bond on a piece of replacement veneer. Some hobby shops sell veneer in different sizes and thicknesses. Some stores like Menards sell veneer. Home Depot of Lowes might sell veneer, but I am not very sure.
If you have a friend with a table saw, a piece of 2 x4 can be cut into various thickness of veneer. If it is a small repair area you can use a popsicle stick or tongue depressor ( a box of 100 depressors costs $1.00 at Dollar General ).
If you go the resin route, try to get some fiberglass cloth. Start cutting the cloth into short small pieces like hair (the smaller the better). Mix these small pieces of fiberglass into your resin. Resin by itself does not have much strength but with glass hairs mixed in it is almost bullet proof.