You want an awful lot in one single trike. On road, off road, and folding. The truth is that you will have to compromise. Narrow, higher pressure tires on a road trike work great on paved roads but suck on a gravel strewn hill. Wider, lower pressure tires on off-road models work better on rough roads and gravel. Expect them to be slower than using road tires on a regular road. I'm strictly an on-road rider and the few times I tried my trikes on gravel were no fun at all but my trikes were made for touring and racing. Trikes that have wider tires and fold tend to be heavier. That matters a lot when you ride a long distance.
I'd suggest going to the Utah Trikes website and looking at the trikes they show there.
www.utahtrikes.com They have all of the best brands and you can look at the various models for Greenspeed, Catrike, ICE, and HP Velotechnic. You can see what they look like and the specifications are there. Most of these brands have some models for off-road. Ditto for folding, but not often folding and off-road in the same model. Be ready for some sticker shock. Lots of people have asked about my trikes but they often shake their head when I tell them what one costs.