Nice!! Won't take much to get it in tip top shape.
Questions for you pros:
1) What model is it? 2000?
I'm with JC and Bman, it looks like a 3000. If you want, pull up the sideburner and remove the 2 screws holding down the control panel. Remove the 3 gas knobs and lift the control panel off. You should find a sticker with the model number and serial number. Should be either on the bottom of the control panel or one of the sides of the frame under the control panel. From there, it will be easy to determine model and year.
2) Is there a specific regulator I can use to replace this one (apparently it's the old style)
Double check where the hose connects to the manifold. It should take a male pipe threaded (MPT) end, which would be
Weber part #7503.
3) Any specific areas I should check during clean up?
Pay real close attention on all sides where the firebox sits in the frame, especially on the left hand side where a bolt goes through the frame into the firebox. This is where rust usually turns up.
4) Will the slats sand up and take a stain/varnish? Any recommendations?
Should have no trouble sanding and staining/varnishing. Don't know much about the process, so can't make any recommendations.
From the pictures, the paint looks pretty good. If there are some small areas that need touching up, give the outside a good cleaning, knock off any flaking paint then just spot spray with some Krylon or Rustoleum high heat paint.
If you want to clean the inside of the firebox, easiest way is with an angle grinder and a 4inch wire cup wheel. Take it down to the aluminum, coat with some PAM and then fire the grill up to season it.