You're going to be in Paris.
From there it's a very fast trainride to Brussels and Amsterdam using the TGV, as well as to many places in France.
Brussels is a great place if you like good food and architecture, just don't try to speak Dutch there (the French speaking Belgians are almost openly hostile towards Dutch speakers).
Amsterdam is nice too for a visit, even if you don't like drugs and sex.
On your Schengen visa you won't be allowed to visit the UK, so taking the Eurostar to London is not an option.
Other major places in Europe take longer to get to than is worth the trouble given the duration of your stay.
Creditcards all over the world place a charge on cash withdrawals, Europe is no exception to that.
You will find that creditcards are less accepted than in the US (or possibly many other places) though. Most stores won't accept them at all or have a rather high lower limit on transactions because of the high cost to them for accepting creditcards.
The main use Europeans make of creditcards is to pay for restaurants and gasoline, and online purchases.
Debitcards are used for everything else and are offered by all banks to accountholders as standard. They're in the process of making them universally acceptable all over the EU right now (they're already accepted by ATMs continentwide).
If your bank card has the Maestro logo on it it will be accepted by most European ATMs, and some paypoints in stores.
As to how much cash to carry, it's hard to tell. As a European I'm so used to paying electronically for everything that costs more than say �10 I hardly carry more than �30 or so.
But for someone who doesn't have that luxury that won't get you very far.
With ATMs readily available though you won't have to carry much.
ATMs will charge you some fee for currency conversion and making the transaction with your bank, just like they would to a European travelling to the US.
But that amount won't be very large, usually something like a few percent with a minimum of a few Euro.
In fact your creditcard company may (depending on your contract) charge you the same and WILL charge you a high fee for cash withdrawals.
Railway tickets can cost anything from a few Euro to 50 Euro or more depending on where you travel to and from (and even when, weekends and offpeak times are usually discounted).
International trains are a lot more expensive. The TGV between Paris and Amsterdam can cost you as much as 200 Euro for a return fare (which is still cheaper than flying though, which can cost as much as 500 Euro for economy class unless you make it a weekend trip).
Museum tickets can be as cheap as free or as expensive as around �25 per person depending on the museum.
Food ranges from �1 for a McDonalds hamburger to hundreds for the best restaurants.
Hotels too come in all shapes and sizes. Count on at least around �50 per night, probably more in Paris (which is known for its hideously expensive hotels with terribly small rooms, some small enough that they use sliding doors because there's no room for normal doors).
In all, count on something like �100 per person per day for running expenses like food, souvenirs, museums, and busses/trams/metro and
you should be reasonably covered (if you choose cheaper restaurants or do as the locals and get some sandwiches in the supermarket to eat in a park for lunch).
That doesn't cover long distance travel and hotels obviously. There especially the sky's the limit.
If you want to visit Amsterdam or Brussels, spend at least a night there and better 2.
The TGV from Paris to Amsterdam takes 4-5 hours, so going back on the same day doesn't make any sense at all.
You might even consider flying out to Paris, taking the train to Amsterdam, and flying home from there. Could be cheaper as well.