http://www.westsussex.gov.uk/ccm/portal/
bunny, a lot of people are now in your situation, and many of them are living out of their cars and vans. In the US a lot of people are doing this, and it is legal to do. I don't know about England. The hardest part is registering for school without an address. You could get a post office address so you have a permanent mailing address, but I don't know if that would fit the requirements for registering for school.
I lived the way you are thinking about for about two years once. I didn't have to. I did it because it was enjoyable (of course, I am a single man, not someone with a son to worry about). I held a steady job, and used my parents address as a mailing address for those years. It is actually a very efficient lifestyle, just not very roomy. I think about putting all my stuff in storage and trying it again for a while.
So, we have to look at the practical aspects of this enterprise. Do you have someplace(s) to park/camp at that is close to the school, or on a bicycle or public trans route? I am surfing the West Sussex website as I type this, so I see that caravaning is very possible there, and indeed encouraged. Whether dogs are allowed in any particular park is something you will have to know. All the details are buried in the website, and you know your way around the area much better than I could, so you should do the research, not me.
As far as your last questions about your benefits and the council tax, you should ask the council about that at: http://www.westsussex.gov.uk/ccm/navigation/your-council/
You can also find more information there about long term camping, and available sites. Good luck. This could be a very fun adventure for the five of you, despite being prompted by hard financial times. It certainly is a beautiful place to camp.