Crazy? Perhaps. First, before passing judgement on your sanity, let's first see if you're simply being honest or not.
What exactly do you mean by "2 weeks"? Does that include the day you arrive or the day you depart? How many FULL days will you have there, not counting your arrival or departure day (no matter what time you think you'll be landing)?
Most people underestimate how zonked-out they will be after flying half-way around the planet and trying to force their body and mind to shift across 9 hours of time zones. Expect your arrival day to be a complete wash-out. The next day - the day AFTER you arrive, your first full day in Europe, you (or someone in your group) will probably still be at half-strength at best.
After you've come to terms with all that, a couple other points to consider: you will surely lose a full day when you zoom between the cities you list. Even with the most efficient transpiration connections, it pretty much uses up a whole day to completely relocate (check out, get to the train station/airport, travel, find your way to your hotel, get lost, find your way, check in, eat a meal, etc.).
While a train makes the most sense between Venice and Rome, the other legs are too long for the train on a trip this short. Fly instead. Either way, each relocation will kill a whole day (the only exception being moving between Rome and Venice, which will only consume MOST of a day).
Night trains? I've tried them, several times, and never slept for more than 5 seconds, despite being exhausted. Some people just do not sleep in something that's moving constantly and vibrating and loud. Maybe you or someone in your group is one of them. I've learned my lesson: night train = no sleep + misery the next couple of days. YMMV.
To your original question, are you crazy? Probably. On a trip this short you could reasonably do Rome + Venice plus one other major city. If you have your heart set on Paris/Barcelona, I'd suggest you pick one of them.