lodging
Most of the tour companies who travel in the park will bring you to their own pre-arranged accommodation, either their own lodge or camp set up. There are a few very scattered ger yurt camps around, but you'll have to check with the park office for their current locations.
Inside the park, there are no commercial lodges, hotels, backpackers, etc. Independent travelers can find lodging with local Khazakh herders if they want to get a very enjoyable and interesting local experience. If you do this, pay them for your stay. Income opportunties for these people are very limited. Two to four dollars a night is currently acceptable and good value for the experience. When traveling further into the mountains, there is no lodging unless you carry a tent, and this is recommended.
There are now several herder communities who have gotten together and can provide some primitive services such as accommodation and guiding. Check with the park information office to find where these are if you want to visit a real community based tourist venture. It can be a great experience, but be prepared for basic, very basic.
camping
If traveling with a tour company they will or should have everything you need for a comfortable camping experience. Take pains to make sure they are reputable and do have what you need. Ask the park information office about good camping areas in the park.
There are very few designated camping sites around the park, so it's basically up to you where you want to put your bones for the night. If around a herder family, it's a good idea to ask if you can camp closer to their ger yurt. Most will enjoy having you there and you will be secure for the night.
The same rules for camping apply here as anywhere else in the world. If you pack it in, pack it out! This is especially important in and around the base camp area for the Tavan Bogd five kings range, where serious high alpine climbing can be had. It is a very fragile area and easily impacted by human use.
AS for camping toilet needs. There are no toilets. If you don't know how to "shit in the woods", you'll want to practice a bit before undertaking a Mongolian adventure.