Hi Hawnted,

Interesting challenge you have ahead of you, seeing Europe in 2-3 weeks. Sorry to read about the reason for it.
As a dutchman living in Switzerland and doing a lot of travel in Germany, I would like to add following to the previous suggestions.
First: look in Europe for your arrangements instead of in the US. Have a look on European web sites, like www.sncf.fr for french trains or www.bahn.de for german trains. There you can get really cheap tickets when booked in advance (TGV: 90 days).
Or www.motel-one.de for really decent hotels in Germany, for example in downtown Berlin from 51,5 Euro (double room)!
A Lonely Planet Guide may also come in handy.
In all these cities, hotels can be very expensive, but there are always very nice little hotels for decent prices. Maybe not a big room like in the Hilton, but often much nicer and more personal. Like in Paris "Hotel Chopin" (around 90 Euro for a double) or Residence Foch (140 Euro).
A good engine for booking hotels in Europe is www.hrs.com (english available). In Germany, autumn is exhibition time. Lots of hotels cost 150% or more compared to normal. Especially in Berlin, Munich and the Frankfurt area.

In Paris: skip the Louvre. The Louvre is very impressive, one of the worlds biggest musea, when not the biggest. You can walk around there for one week without seeing one thing twice. This is frustrating when you only have a few days. Instead, see the city, the people, which is one of the best things of Paris. Rent bikes. Climb the Eiffel tower. When you like Monet, go to the Orangerie and the small hidden museum "Marmottan". Otherwise see the Musee d'Orsay. There is a very nice restaurant called "Bouillon Racine". They have excellent 3 course menus for 30 Euros, which is cheap in Paris.
Train travel from Paris to Amsterdam: yes, good idea.
When in A´dam, ride bikes. But: see to it that you get out of the city too. See the Ijsselmeer (part of the sea that was dammed and now is a big lake) when you like sailing; go to Den Haag when you like art (also some good musea in A´Dam though). Go to a place by the sea in the dunes (but not Zandvoort, Scheveningen is better (tramline 1 from Den Haag central station).
Berlin has a lot of art and history and is relatively cheap. I would definitely go there. See an opera if you like it, the remainders of the wall, etc.
When you want to see the mountains, go to Switzerland. Otherwise: skip Switzerland.
Train travel from Berlin to Switzerland: I´d rather not. Take the plane from TXL to ZRH. This will save a lot of time and with AirBerlin this will cost you only $66 per person. And: AirBerlin is not a lowcost airline like RyanAir or EasyJet, so no funny stuff with extra charges for printed boarding passes or fast boarding or luggage surcharges.
When going to Switzerland, take the train from Zürich Airport to Bern (1,25 Hours). From Bern you have easy access to the mountains. For instance, you can go on a via ferrata in Kandersteg, which is very impressive. You will never forget that. If you want to go up to a peak and not climb youself, go to the top of Europe: http://www.jungfraubahn.ch/en/DesktopDefault.aspx
From Bern you can take the train to Milano, from where you can go on (by train or plane) to Rome or Venice. Trains in Italy run also in the night, bur to Rome is quite a distance.
When you like to see southern Germany instead of Switzerland, go to Füssen if you like castles (you can also do this on a day trip from Munich). Near Füssen, you will find Schloss Neuschwanstein (http://www.neuschwanstein.de/), which is without doubt one of the most impressive castles in the world. In the village nearby you can relax in hot springs (Kristalltherme; http://www.kristalltherme-schwangau.de/). I believe you can book both as a package.
Venice is nice to see, but be very selective and don´t spend too much time there. Besides: it is probably inaccessible in autumn due to flooding (check in advance). I would rather concentrate on Rome when in Italy. Be careful for pickpockets there!! And: when looking for Espresso, you can watch the price rise by the meter when nearing tourist attractions. Near big sites, an espresso can cost 5 Euro or more (normal price around 1 Euro).

Hope this helps a bit rather than confuses (all the possibilities). Otherwise, just ask

SuperVoid