Here is my first question, is MDF stable enough to allow me to attach a thin piece of cedar to one side and a thin piece of oak on the other, by thin I mean a quarter inch for example
I'm convinced that some of the great cabinetmakers of the past would have near as darn it killed to get hold of MDF. It is extremely
flat (much better than any plywood), very stable in interior locations
, it is highly consistent in structure and it glues well. That makes it an ideal substrate for commercial veneering and is indeed why a lot of commercial furniture such as office reception desks, fittings for high-class pubs, restaurants, hotels, etc are made from it. But note that I said commercial
veneer, which is typically 0.6 to 1.5mm thick. You are absolutely right to assume that whatever you put on as a face veneer will need a backer veneer on the opposite surface - miss the backer and the veneer will pull the entire panel out of flat surprisingly quickly - but when that veneer gets beyond a certain thickness, say 2 to 2.5mm it starts to take on some of the characteristics of the solid wood it was created from. 1/4in (6.35mm) is a lot thicker than the veneers used in plywood. That means that it will move with changes in the air temperature/humidity - and it will move at a different rate from the MDF to which it is attached. This will result in all sorts of problems of veneer cracking, detaching from the substrate and panels warping - it may take a few years, but it will happen. For that reason many professional cabinetmakers avoid using overly thick veneers and stick to the commercial stuff for larger areas.
On the subject of laying down veneers there is probably no easier to learn technique than hammer veneering (using a home-made veneer hammer and hide glue). Ellmers sell a cold hide glue which is ideal for veneer repairs on antique furniture and for veneering small pieces. It has the major plus point of being completely reversible even after it has set - just warm the veneer with a dry iron and the glue bond will "fail" and can be altered, adjusted, etc
If you are interested in vacuum veneering ther is an excellent site on the subject called Joe Woodworker
which covers the subject quite well, including detailed instructions on how to build your own low cost vacuum veneering press