1. May is not a peak season and so the flight ticket may not be too expensive. Your budget can be enough if you do not need to eat very well and stay in hostel / 2 star hotel.
2. You can walk-in for some hostels but of course booking in advance is better.
3. I suggest you to buy eurail ticket as it is very convenient to travel to any cities you like, provided that you got the appropriate eurail ticket. There are many types of ticket and you can buy the most economic / suitable one for your trip.
4. 15 days is not too much for such 3 big countries. I think you can spend equal time for these countries since each has many things to see. For UK, London must go. For France, Paris must go.For Germany, Berlin must go. But it depends on the route you travel and may have to change to other cities.
5. Not too dangerous if you are alert at all times and avoid going out late at night.