1) What Emery said. Ideally a piece of 3/4" thick hardwood ply because you'll want to drill holes for where the mount holes are on the pedal board (under where your heels sit) then have bolts going UP through the board. If you do it like that without any more effort though you'll have the bolt heads trashing the floor so you'll either want to add rubber feet (meh) or better, drill out a larger hole about half way through the wood for the nut head to settle into. You'll still want to put something on the bottom of the wood to create some friction and so it doesn't trash the hardwood floor but if you counter sink the bolts height isn't really important. I ran these pedals with just these 2 bolts without any issue. I only added bolts through the actual individual pedal assemblies when I added tactile transducers because they vibrated, but the setup is solid with just that bolt. If you get creative you could add a block to the back of the wood to line up / lock in with part of your chair for added stability to lock it all in place, or you could just put the wood under the carpet, and the chair on top like you do now if that's not slipping. If you have to put this away when you're not playing you could hang it all on the wall. Advantage is if you can integrate chair mount onto this then it's really stable.
2) More compact and this is what I would do because the wall is right there. Create a wooden L shape platform that the pedals are mounted to with countersunk holes again like above. Use triangles of wood on the sides or struts so it's really rigid, because the L not collapsing on itself is what makes this work. Put a wooden strip across the back at the top that's the same thickness as the wooden trim on your wall so when pushed against the wall the back board is upright and solid. Make sure that it's high enough that there's a gap between the wall trim and the cross board for those cables to go through. The risk is that there's nothing there but drywall which could be a problem. You could just push it in
so make the wood across the back is wider than the rest of the build. There will definitely be a stud at the corner on the right, and should be one to the left at 18"? but if you tap the wall and check for something solid (or use a stud finder), then go that far that would be ideal. See crappy drawing attached. You could alternatively do this with two strong L shaped steel brackets attached to the pedals, and the same kind of wooden strut across the back for the same purpose. In terms of height, if the point of upper contact with the wall is higher than the vector that you are exerting on the pedals then it won't pivot up at your heels but I'd go to the height of the pedals + 2 inches just because. If you REALLY want it to be secure, and don't care if your pedals are always out, screw through the cross piece at the top into studs. Hell of a lot easier to patch drywall when you move out than hardwood. In terms of materials you could build the L and sides again with 3/4" hardwood ply, or even from the 18" pine project boards you can buy. The triangles at the sides make it really solid. The strip across the back could be a strip of pine because the load is so evenly spread across the wall.. it's not going to flex.
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EDIT:: Would be stronger if the triangle was INSIDE the L like below, not screwed to the SIDE of the L like above. Now strength is in the wood, not the screws holding the wood. You could also use metal brackets to enforce the L as long as they are strong enough.