No, you can't go inside the cliff dwelling.

I was there in the mid to late afternoon and thought it was perfect.

I don't think so. It is viewable from the parking area, but not a lot. It can be a very short walk to look up close.

You could drive there in your own vehicle and do a self guided tour.

You sound remarkably organised - we just made a list of attractions and did as many as we could - as to what’s covered I have no idea, either go back to the website or call them.

In your situation, I would not visit Montezuma Well due to lack of shade but I would highly recommend visiting Montezuma Castle because it is truly incredible and there is shade with benches to view the ruin itself. It is a very short walk from the visitor center to the ‘castle’ ruin. Do it!

Get a National Parks Pass for $80 for the year (or less if you are a senior, or free if you are military or have a 4th Grader with you - make sure you get the annual pass for them.)

Once you get yourself parked you'll be within an easy walk to the entrance area. After paying for admittance and exiting the entrance building it is an EASY walk to view the Monument, at very most 5 minutes. There is a short and scenic loop that will take you close to a small river and a "Diorama" that will show what is behind the Monument facade. A short, mainly level and easy walk.

I spent between 1-2 hours there (I also like the indoor exhibits). We did not have chance to visit Tuzigoot.

No. There is a gently slipping walkway that would be appropriate for a walker though. The "ruins" are not extensive and the path is quite short.