1. Nothing much, but do wipe it off as best you can as soon as you can. I did some experimenting with AQ on both silver and brass a while back - it will eventually color/etch the metals, but it does not happen instantly so you have plenty of time to wipe the inlays. Hit those inlays with the heat source first, just a drying heat.
2. Unless the stock has shown a tendency to readily twist or warp with the barrel out of it then you are good to go for the length of time your finish is being applied. I would put it back in during those long drying times if you are using an oil type finish. Wait until most of the solvents have evaporated before doing so.
dave