I am planning on a circumnavigation of Iceland this summer and the Ring Road might be the only road available for lenghthy stretches. I would like to know if camping along the Ring Road is permissable, especially once I get out into the more remote areas. I would also like to know if ground/river water is safe to drink without purifying.
Wild camping is permissible as long as you maintain a respectful distance from homes etc, but in some places it might difficult to find good spots to put up a tent close to the road. Based on my experience, the best spots are found taking side roads a little bit off the main ring road. Near populated areas you can usually find official camping sites (check the link provided by MeNoTalk).
The water from small creeks/rivers is almost always safe to drink without purification as long as you avoid getting it downstream from habitations.
Along the Ring Road it's acutally harder to NOT find camping areas than to find one. They are literally all over the place. Prices and facilities can vary greatly. On some grounds you get free internet and hot showers are included as well. On other grounds (the more crowded ones like Skaftafell, the main camping ground at Vatnajökull) you even have to pay to recharge your cell phone.
But, they all offer toilets, which is why I wouldn't bother with wild camping if you stay near the Ring Road anyway. As for wild camping, it is allowed to some extend, as is explained here:
If you wish to camp on cultivated land or near residential buildings, fenced off farmland, or such, you have to ask permission from a landowner or other beneficiary before you pitch up the camping tent. The same rule goes if you intend to stay longer than one night.
If there is a camping ground nearby, you have to use one of those. I don't know what exactly is considered to be "nearby" though. For the sake of preserving the beautiful and very fragile landscape I strongly advise against wild camping in the more crowded areas.
Also, the camping grounds all have running water which is perfectly save to drink (even though it may smell funny). To save you from unneccesary trouble, I suggest you just stay at a camping ground.
Usually the price for one tent is around 8 ISK per night, Camper trailers are more expensive, but I don't know the price for those.