I have never seen a good folding table for use as a cutting table. It is possible, I suppose, that you get a high end office table that is technically portable that could work. But the type you get at Costco even? Not stable enough in my opinion. And certainly not the type of one that you can get a tent to fit over it to amuse the kids. For something new, affordable and possible you might look at computer workstation desks that require assembly. Check them out for flimsy-ness in person.
I'd be looking for a piece of furniture the right size with good lines but probably a bad finish at the thrift store. Some times there is a lovely piece of maple or something under that bad paint job and stickers... and even if not a quick and dirty refinish on the legs on a piece with good lines can be just fine, and you want something about the size the extra large mats cover and/or don't want to have to worry about the top surface anyway.
Personally, the best height for me to cut at is standing and dining room table height. I have another friend who vastly improved her cutting by standing at kitchen counter height. In a multi-use room, I'd consider building in a counter the height you like or with storage underneath or maybe standard counter height for part and desk height/open underneath for part.
I have a tolerant hubby and I've deliberately spread out my sewing so I get more steps in during the day. I also used to do all of one set at a time, so if that was 200, so ("sew") be it. Now I do groups more like 20-24 at a time, get up and iron in the living room, cut in the dining room, go back to the back bedroom where I layout stuff on the bed (house too small for a design wall), and then back to the sewing room. It may not seem like much but its actually made a huge difference in my previous more sedentary life style.