1st workbench

-Many (good) workbenches have been made from solid core doors.
-The humidity would likely have minimal effect on the door. No worse than a solid wood bench. I wouldn't go to any great lengths to mitigate it.
- Building a skirt out of the white oak is a good idea. I'd drill countersunk holes every 8-12" in the oak, then glue/screw it onto the door. Maybe drill out the screw holes deep enough that you could cover the screw heads with dowels (then cut the dowels off flush with the skirt).

I had one of those benches for years mine just set on saw horses and, it moved with me over the yrs from shop to shop until I built a roubo bench for my permanent shop I build 6yrs ago. I never did anything to the solid core door for stiffening, I did add a cheap vise I ended up screwing some plywood to the top once I beat the crap out of the veneer on the door.

A skirt would add a lot of rigidity to the top but you might want to consider how you are going to use the bench and if it will affect that. For instance a 5/4 skirt might not be thick enough to register a quick clamp against straight on so you might want to add a filler piece behind the skirt to thicken it up and make clamping things to the top easier.

For me the most important part of a workbench is it's base. I hate workbenches that rock at all and find them very hard to work on. My current "temporary" bench is a bunch of 4X6's glued together with a splayed leg base that is rock solid. The bench is ugly and not as flat and smooth as I would like it to be but it's solid which goes a long ways to make up for it's shortcomings.