Your LS1 brakes might fit w/o doing anything to the shocks, so I'd try that first.
Mounting both shocks to the front side of the axle is much more involved that just using a new lower plate. You would also have to change the upper shock mount, which is part of the unibody. (Unless somebody makes a "S" shaped shock absorber
) Relocating the upper mount could be done, but would require some cutting, welding and engineering to make sure it is properly supported. Look in your trunk at the upper shock mounts and you'll get a clearer picture of what's involved.
The shock relocation kits I've seen are nothing more than a round or square tube bolted/welded between the frame rails with upper shock mount tabs on them. Pretty simple, but as I said, use shorter shocks, and mount them to the inside of the leaf spring which makes for tons of room for the caliper. I looked at doing that on my car, but I would have had to redo the tail pipes and didn't want to mess with it.
As far as not liking the staggered calipers, that's a matter of preference I suppose. From a "looks" standpoint, you can only see one side of the car at a time. From an engineering standpoint, it makes no difference. Look at some OEM set ups and you'll see rear calipers in front, behind and at the top of the axle (JL8) The rotor couldn't care less where it's being pinched from. The "big boys" who make rear kits for staggered shock cars where the caliper and the shock won't fit on the same side of the axle do the same thing. (Baer, Wilwood and Kore3 come to mind) Baer and Kore3 do it with PBR calipers.
On the wheel hop thing - I don't know. I think GM used the traction bar on 67's partly because of the non staggered shocks and partly because 67's only came with mono leaf springs.
Good luck with whatever you decide.