Wouldn't want to ride a bike (bicycle) up from the Morefield Campground Area & even less to ride one from the entrance to Far View (visitor center) Area. Maybe ok from Far View. Roads are a little narrow to be sharing the roadway on a bike with cars & RVs. It a motorcycle or scooter (bike) wouldn't be too bad. I'd ridden a 250cc enduro from the CG to Far View.
There are fairly large parking areas at Far View. One could park there & bike to the other areas. When we were there years ago, people would catch a shuttle bus to go see one area of the park, the Wetherill Mesa area.
Likewise, you should be able to find parking farther in at the Spruce Tree Terrace area & might be able to use a bike around that area. Large Tour Buses drive up to that area (not shuttle ones).
From what I remember, the drive from Far View to Spruce Tree area is pretty level going. But it has been since 2010 since I've been there. We only did a drive in & overnight on that trip. There were a few wild mustangs along that way that evening. Loose Horse anyway.
You could ask a Ranger about taking the RV on the Mesa Top Ruins/Cliff Palace Loop Rd.
All depends on how proficient one is on a bike (peddle one), to be able to use one on the Loop Rd. But my memory of that road is a little vague, not real sure if has descents & assents or not. Maybe 1997 since I was on it.
:CDesertHawk- Las Cruces, NM USA
*2015 Lance 1985~Casita de Campo~23' 4" Tongue to Bumper, Dinette Slide
160 watt Solar Panel/GoPower! Solar Controller
*2009 White Ford F-150 Reg. Cab
Long Bed with A.R.E. Molded Fiberglass Topper
*Previously~ 2005 16' Scamp