I'm not a expert in this field but i can give you some ans to some questions. You need to figure about 200-300fps less than what a rifle will get in your pistol caliber. I dont think the 7wssm would be the best for this rig, more like a 7wsm or 7 Dakota
. 18" barrel to give better results. You will have to go with a custom rig or some kind of re build to get what your looking for. I would go with a custom and get exactly what you want the first time. Depending on barrel contour is where the weight will vary. I'm sure some of the others will chime in on what their rigs weigh.
My 30BTAC varmint pistol is 14# with a Nightforce 8-32x scope, so that ans your other question,
yes the 5-22 would be a fine scope for your build. My pistol is set up for bench or prone shooting, not carrying around the woods. I have a 20" barrel on mine.