This is a very good question and the answer is given below.
There are two base camps for Mount Everest.
Nepal side or South Side - 5,364 metres (17,598 ft)
The one that is widely used and visited is the one from Nepal and that is the one under discussion in this topic. BTW this trek is the most visited treks for any mountain.
Tibet / North side
The other one is accessible from Tibet but is not visited much due to the rift between China and the west over Tibet. It is accessible by road and a few hours of trek. There is a Tibet permit required to visit this area on top of the Chinese visa. It can be visited by traveling from Nepal to Tibet or from going to any Chinese big city and then travel to Lhasa by Train, road or by air.
Regarding your question of faces of Everest. Everest is not as lucky as some other 8000ers as it is hidden behind other giants and only a tip is visible from even closer proximity. The best view is from the top of Kala Pathar mountain but as I mentioned still the tip would be visible but with a little extra.
It is hiding behind Lhotse and Nuptse and I'll share pictures of this formation as we reach closer to Everest. Lhotse (8,516 metres (27,940 ft)) is a minor 8000er as it is connected to Everest via the south column and has only 610 meters climb left to its major top although it has three peaks all over 8000.
I hope this information will be helpful.
