EBC basically has two main seasons: Oct-Nov and March-April. Oct-Nov is by far the most popular because the skies tend to be clear and weather is good, but the crowds can be horrendous. The March-April season is more quiet, but there can still be some snow and the sky starts getting more hazy in April.
I hiked it in 2 March- 14 April 2014, in and out from Jiri (can be reached by bus from KTM). It was fantastic! I reached EBC on the 25th March or so and there were already some sherpas setting up tents for the Everest climbing season.
Regarding your questions.
1) Leave your tent at home. If lodges are full you can always sleep somewhere on the floor.
2) You get your water from the teahouses. It does need purification. I carried a Steripen Classic (UV) for this. I have never carried more than 2 litres of water during my hike, but of course it was still winter so you may need more if you hike in October.
3) It is very easy to do it independently and solo. It's an inhabited area, not a wilderness trek and there are plenty of people around. Just follow the crowds. When is doubt, you just ask the way (I did this a few times on the Jiri - Lukla part). Only exception is when you go beyond the route that you posted, across Lhenjo La, Tsho La passes etc, where I didn't see many people when I crossed (early spring). If you only hike Lukla - EBC, that is basically an broad, crowded trail. You cannot get lost really.
Food is nice in the lodges. Nowhere did I have to pay more than US$1 for accommodation (private room!). On the Jiri-Lukla part I was often the only foreigner in town, but Lukla-EBC was very crowded even in March. Always got a private room though. They make the money from the food, but even that was cheap (but way more expensive than KTM). Prices go up the closer you are to EBC, but the most I ever paid was US$6 for Dal bhat (big filling meal). On the Jiri to Lukla stretch it was more like half that price. Of course that was in 2014, so it could be different now.
I have an extensive walk report of this in the link in my signature.