I think you have the start of a wonderful trip. I've done a similar trip in the past.
Fly to Heathrow, bus to Oxford.
3 nights in Oxford. Oxford is a great place to get over jet lag. Do one of the walks offered by the TI, take the bus to Blenheim palace, rent a punt along the river, visit a museum or two, watch the college kids zooming around on their bikes.
Rent a car for the Cotswolds or take a train to Moreton-in-Marsh and then use taxis. (We used taxis, not as convenient as a car, but doable if you do not want to drive.) Chipping Campden is another town you might want to stay in. You can take several walks, visit Hidcote and/or Kiftsgate gardens.
Personally, I'd skip Stratford and spend my money on tickets to a production at the Globe in London.
Take the train to Bath and spend a couple of nights here. It's a great walking city with lots of history.
Take the train to London for final days. London is expensive, but you really should stay in central London to make the most of your time there. Remember that all of the major museums are free, which helps with costs. There are lots of easy day trips by train out of London.
Fly out of Heathrow.

Forgot to mention, I really like the Buttery Hotel in Oxford. Reasonably priced, very central, includes great breakfast.

Going back to your question about accommodation in London, I have stayed at a couple of the LSE halls of residence (High Holborn and Northumberland House) and they are fine. While I love the Hampstead area, for a first visit I would stay in a more central location.

If you decide to spend a night in Oxford you can sleep in one of the colleges: http://www.universityrooms.com/en/city/oxford/home

If I were touring the Cotswolds and had a car I would consider staying in one of the National Trust properties: http://www.nationaltrustcottages.co.uk/