Comments on the recommended routes:
I-70 vs. US 285:
I-70 can be quite busy. Avoid leaving Denver any time after 4:00 p.m. and skip altogether Friday through Sunday. Tunnel is at 11,000'. Fremont Pass is at 11,300'. The climb from Copper Mountain to Fremont Pass has a passing lane virtually the whole way. Quite scenic, but you have to go over two significant passes.
US 285 can also be quite busy on weekends, but the traffic thins out after Bailey. Kenosha Pass is under 10000' and a much easier grade than the tunnel approach on I-70. Trout Creek Pass, just before Buena Vista, is almost imperceptible coming from the east.
You'll see more higher mountains on the I-70 route, but the scenery on US 285 isn't bad either. South Park is a huge intermontane basin best viewed from the top of Kenosha Pass.
Cottonwood Pass vs. Monarch Pass:
Cottonwood Pass is 800' higher (12,100') than Monarch. Far more switchbacks also. It is a bit shorter distance, but longer drive time.
Both passes have great views from the top. Traffic is easier to deal with on Monarch as there are fewer curves and more passing lanes. Also, as Cottonwood Pass was only recently paved all the way over the pass, it has become a very popular and busy route. I somewhat prefer the scenery from the top of Cottonwood Pass to Gunnison over that from the top of Monarch Pass into Gunnison.
No matter which route you select, one somewhat obscure stop worth taking is at Cimarron. Turn right where the sign says Train Exhibit/Morrow Point Dam. It's just over a mile drive (paved road) back to the dam. You'll essentially find yourself at the bottom of the Black Canyon of the Gunnison although further upstream than the NM. Kind of a different perspective. If you're into old narrow gauge railroad cars and engine, stop and poke around at the Visitors Center. I don't think they open the building anymore, but the outside exhibits are open. Nice CG there for an overnight. We've been there a number of times. Rarely fills, or if it does, it's not until the evening.
Yes, you can drive on the South Rim Road with your rig without issues.