How does the training for an EASA license from the US compare to that from Europe?
You definitely do not need to answer any theoretical questions before beginning flight training in Europe. In fact pop down to your nearest airfield, find a flight school, and chances are they will get you in the air (even for a trial flight) as soon as they are able. 

Whoever told you that may have been thinking of the examinations you need to pass to get your PPL, and while 1000 questions is probably a high estimate, you need to be able to answer exam questions on a bunch of subjects such as Meterology, Air Law, Human factors and radiotelephony. You do this during your flight training, but again there is none of them you need to pass before you can begin.

I am from the UK, and I learned to fly in California. It was before the EASA licenses (In those days they were JAR licenses) but the idea was the same - I was training in the US for a license that was valid in Europe. I had to do the same training, and take the same exams as if I was in Europe learning  there was no difference.

On the cost, yes it was slightly cheaper per hour in the US however when you factored in getting there, staying in accommodation and food etc I don't think there was really that much in it. It was more an opportunity (excuse!) to take a 4 week holiday and do nothing but fly every day in great weather (Something which in the UK, at least, is hard to get to do).

Fun fact: I did this 11th October 2001. Conversation at LAX with TSA having explained I'm entering the US as a lone male to learn to fly - "Don't worry Mr C, you don't fit the demographic we're currently looking for".