As @mstrauss has said, solid wood is probably not a good idea unless you live in a consistently dry climate.
However, what you use for the electronics enclosure probably doesn't matter much.
If you are a woodworker I have a suggestion for the corners, braces, etc.- Use 3/16" aluminum plate covered on its faces with thin sealed wood veneer.
Scuff up the aluminum faces with 80 grit sandpaper and use a good contact cement.
The veneer should adhere strongly.
That way you would get the look of wood with the stability of aluminum and could polish up the edges to a gloss.
I think it would look fabulous and should not be all that difficult to do.
I too have done a lot of work on aluminum parts using woodworking tools.
Just feed very slowly and smoothly and wear ear, eye and face protection !!!!
As for the PSU, yes you should inspect and correct any deficient solder joints you find.
Its case is part of the heat sinking system for some of the components, so only leave the cover off if you want to help airflow. Keep the base chassis in place.
Better yet, change to a mains-powered bed heater and a 24 volt PSU as so many others have done (including me).
Heat is one of the biggest enemies of electronics in general.
The completely pitiful enclosure fan and inadequate vent holes are a major issue on these printers IMHO.
I added an 80mm fan into the rear cover to blow fresh air directly at the logic board and an 80mm exhaust grille over the PSU.
They made a world of difference and were well worth the effort.
Fans generally don't take much power (unless they are very large) so 26 or 28 gauge wire is typically more than adequate for them.
Some approximate rules of thumb for common wire sizes are- (copper wire)
It's all about the current draw of the loading device, how much voltage drop you are willing to tolerate over the length, and how hot the wires can be allowed to get.
So unfortunately there are no hard and fast answers.
As for lengths, whatever works basically.
Leave service loops at the ends if possible.
For routings, keep them up against framing members and out of harm's way.
Clip or tie-wrap them to keep them secure, but NOT inside the cable chain (they will become stressed and break quickly).
I hope this helps.