Depends how you want to travel and how much you want to see, do you like nice hotels with a buffet breakfast, lunch and dinner or are you happy with a room with a bed and not much else because you know you'll be out all day and partying all night?
$3500 is about £2150 or 2600Euros, from the look of the PDF it looks like you're stopping at a couple of places in each country which could be done cheaper doing it on your own by a fair bit.
Considering exchange rate, London will be your most expensive city. Think of costs here as being without the currency symbol in front, so if a coke costs you $1 there it will be £1 here from a kwikemart/corner store which will actually be $1.6 to you.
I'd get your flight to london and stay in a hotel (travelodge or similar) in zone 2 or 3 (london tube zones go from 1-6, 1 being the centre) for 2 days, hop on the eurostar train to paris, spend a couple of days there and get a eurotrain card which allows you unlimited travel over the train network in europe. Try to ensure that any travelling is done at night to save on a hotel and pack light.
In may it should be quite warm/hot here. If you wanted your trip to last even longer/cheaper it would be best to go direct from London to eastern europe (estonia, latvia, czech etc) where costs are cheaper.
live in London, have travelled around europe for a year, 2 years in southeast asia and india and next trip it to south america in 2 years!!