Hostel Backpackers

As you have not given details about your plans as they stand now, I can only give you general information.

If you travel in July or August you will need to make reservations, but even then you can do that by phone the night before you arrive in a town if you are flexible, September and later in the year you will still want to phone in if you arrive late in the day, but if you arrive early there is no need for that.

Costs:
Western Europe, hostels, doing your own cooking, spending little or nothing on transport and drinking, about €50 per person per day. If two share a cheap hotel room and are careful in eating out, €60 to €75.
I budget €100 for myself nowadays, and if you can, you should do too, it is so much more fun if you do have enough money to do a bit of a luxery thing every now and again.

For London and Paris, ideally a week, but at least 3 full days for sightseeing, if it is the first stop on your travels, add one more day just to get everything arranged.
Other capital cities in Europe at least 2 full days, ideally no less than 3. And for each capital city, also visit a smaller town and a bit of countryside, a national park or beach resort, so you get a better taste of the country. I would say no less than 2 weeks for one country and at least one week for all the others, but knowing your short American vacations, that might be asking too much.

If you are students or young enough to get student ID’s without acutally being a student, get that ID.
Also get an Hosteling International card, specially if you intend to stay in HI hostels. But if you doubt you will use them, you can still stay in them, you will pay a surcharge for the first few nights and with that you will pay for your membership, After a few nights you will be a member.

As I do not know your schedule I can not even guess whether a rail pass is better or not, but in general, on a 10 or 15 days pass you need to be on the train all day every day your pass is valid to make a profit, that is doing 10 or 15 long distance or overnight journeys, all using ‘normal’ not high speed trains.
If you want to use the high speed trains often, you will mostly have to pay big surcharges, and have to make seat reservations. If you use the ‘normal’ trains you can just hop on and off as long as your pass is valid.
This site has good information on rail passes and alternative ways to travel:

Added after finding your travel plans:
I would stay one day longer in London, (cut it off Amsterdam)
Take a day off Hamburg and one off Prague and use those two days for a stay in a small town, either in Germany or in Czech republic, use your Munich days in the countryside instead of in town, (or do that for your Hamburg days.)
If you loose most of a day for your Rome to Paris travel, take it off Rome, not off Paris.
And instead of returning to London, stay in the English countryside, so you have seen that too.

Travels to and from London can not be done on your rail pass, better book your tickets as soon as you know when you will travel / or when the dates for that day come available, I think the site is called Eurostar.com or .co.uk, you will find it with google.
That leaves you with 6 (your plans) or 8, (my plans for you) days travel, so a 10 days pass would work, and with these distances I guess it will work out cheaper than just buying ticket.