For example, what would the rotation speed (Vr) and stall speed (Vs) be at maximum takeoff weight versus flying with a single occupant? (2297 lb vs 1807 lb)
Instead of doing detailed calculations like you were in a jet, there is a useful rule of thumb advocated by author/instructor William Kurshner for working that out, which while not exact down to a fraction of a knot, is conservative and easy to do and is therefore perfect for regular GA flying.
Reduce Vs/Vr by half of the percentage of the weight reduction. So if stall speed is 50kt at gross, and the weight reduction from 2297 to 1807lb is about 21%, and half of that we can round up to about 11%, your stall speed is going to be about 45 kt at that weight.
So if I was using say, 55kt as a rotation speed at gross (I seem to remember that's what it was on the 172), I'd use about 49 kt at 1807 lb. Close enough for 99.9999999999% of normal operation in a Chevy Biscayne Of The Air like the 172.
The THEORETICAL way to calculate reduction in stall speed is to take the listed stall speed for 2297 lbs and apply math as follows:
square root (1807 lbs/2297 lbs) x 50 knots. Comes out pretty close to what John K says.
The SAFE way is to take your plane about 3000 feet AGL in the flap configuration you choose and actually do it, making sure your eyes in your plane at your weight gives your actual stall speed on your airspeed indicator. 172s are very forgiving, just make sure CG is right and stay on the rudder; do these all day long, power on AND power off.
As far as reduction in rotation speed, this may have an application for an emergency backwoods muddy field takeoff, but little else. The "trike" gear is designed to keep the plane below the AOA needed to fly until you rotate. In addition to being safer, it also shortens takeoff roll by reducing drag.
Rotating at a lower speed increases risk of a freak gust lifting and slamming the plane back down, and lowers safety margins should cross winds require any type of aileron or rudder input. It is especially a bad idea in higher performance aircraft, where an uncontrollable torque roll may develop. It might be best, especially on a windy day, to stick with 55 knot rotation speed. It is a good habit to be in.