Are tourists still allowed to enter the Great Pyramid? Is access still restricted to the King's Chamber? Is there any way to "upgrade" your experience and get into the other chambers?

In Cairo, what are the major viewable sites besides the museum of antiquities?

Thanks in advance.

Yes,the great pyramid is still open to go inside. I've actually never been inside that one, but everything that has been typically open in the past is still open inside, I'm quite certain.
We find however, that most of our clients are happy to go inside the Red Pyramid of Dashure first and often decide after visiting that one to skip the Giza pyramids as they are not as impressive in general, plus they cost more to go in. The Red Pyramid is Free. The Khufu Pyramid costs LE100.

In Cairo there are SO MANY viewable sites. At minimum you need to visit Dashure, Sakkara, Giza and the museum for our antiquities, but if you have time there are more sites to visit, in fact you could spend an entire day at Sakkara with all that is open out there.
If you are interested in Christian and Islamic history, we could show you many many many churches, monasteries, and mosques all close to the city.

If you are interested in Cultural tours, there are so many of those that are offered - Crafts, cooking, dancing and more.

Oh, I forgot to tell you about the REAL antiquities here - how about a day to go see The Valley of the Whales? Wadi el Hitan. These make the pyramids look NEW.

What do you think of the Citadel and Cairo Tower?

Citadel is fabulous IF you work with a company that will show you something more than just the Mohamed Ali Mosque. There is lots more there to see than just that mosque.
There are loads of examples of old Islamic art and craftsmanship if you know where to look.
The Mohamed Ali Mosque is a good mosque but it is new compared to the others there, and if you've seen the Blue Mosque in Istanbul, then you've seen something real similar. This one was built from the same blueprints.

The Cairo Tower however, is not my favorite site to see, but having said that, IF you have an afternoon of very clear weather, it might be worth going up for photographs from the observation deck. DO NOT bother eating there though.
I heard a rumor (and believe me - until something actually happens around here, it must be considered a rumor)that President Morsy was going to sell the Cairo Tower to some other country to run.
If this happens, I suppose it is possible they might actually improve it. Right now, it makes for a nice observation point IF the weather is decent, but otherwise it is kind of useless and not all that nice.
Back in the old days (5 years or so ago) I took people up into the Hyatt Hotel bar on the 40th floor for essentially the same view. The Hyatt did not offer a 360 view, but a wonderful 180 looking out to the pyramids and the Nile. Considering the elevator in the Hyatt felt safer and it was free to go up in it compared to paying to enter the Cairo Tower, it was a find. Unfortunately the Hyatt has pulled out and I'm not even sure it's open these days.

Unfortunately Kaye047, the only Nile Cruises going out of Cairo would be with Ambercrombie & Kent. If someone is interested in that, check that out, but to the best of my knowledge, they are not doing these yet on a regular basis.

Maybe I misunderstood, and perhaps you were really talking about a private yacht cruise around the city or a felluca sail boat ride instead? Either of these are interesting too.

As for the monasteries you mentioned, the only ones close enough to Cairo to do from Cairo are the Wadi el Natrun group. The others are better done from a base on the Red Sea, either Hurghada or Sharm/Dahab.