You do unweight the saddle by putting weight on the pedals – we also tend to run very low tyre pressure in fattish tyres, which takes quite a bit of the sting out (I have ~20psi in a 26×2.4, though some run as little as 15psi in 3″ tyres). The saddles are also quite padded compared to what you use on a bike. But yes, there is still an issue with your plums, though given you have issues with those from when you first start riding on the flat it’s something you learn to cope with by hitching them out of the way as much as possible.
The lever for the brake is under the front of the saddle – here’s a disc brake uni where you can see that setup. However you’ll see they’re grabbing the front of the seat even when not braking – that’s because there’s a handle there which has a variety of uses – it helps to control the uni, you pull on it to help you stay on by increasing the foot pressure on the pedals so they don’t bounce off (and to stop you losing contact when catching a bit of air), you pull on it to give you extra force on the pedals when climbing like standing up on a bike, and you can also use it to take a bit of weight off the seat to ease your contact area, or for repositioning stuff. If you get into doing hoppy trials type stuff (which extreme off-roading does start borrowing from) then it becomes even more useful.
Very impressive riding and very inspiring, and I don’t want to detract from it at all, or suggest that I’d ever ride stuff like that as I probably won’t, but in a sense it’s actually easier than it looks (once you get past the point that riding a uni at all takes a lot of practice). After only a year of riding, stuff like that doesn’t seem totally impossible, and certainly some of the easier stuff in that I would ride. Despite being very much a wheels on the ground biker, taking a bit of air on a uni no longer really bothers me at all – if anything I seek it out – as you can see from that video, when it all goes wrong you just get rid of the uni and run it out rather than getting slammed into the ground as you do on a bike.