My family and I rode Charleston WV-Chicago IL-Red Wing MN-Glacier Park MT-Portland OR in July '04. In this case, "family" means me, my wife, 3 kids ages (at the time) 3+, 6+ and 7.5, plus two sets of grandparents. We broke the trip up--1 night in Red Wing MN, 2 nights in Glacier Park--so no more than one night in a row on the train. We flew back home.
Onboard, we had the advantage of granparents willing to spring for deluxe sleepers, so we had the luxury of opening up the doors between bedrooms during daytime travel--very nice. Enough space for all 9 of us, if we wanted to, though more typically it was 6-8 in the combined space. In general, though, our kids are old enough that we try to get one standard sleeper and one deluxe sleeper on our overnight trips--the deluxe because the lower bed is wide enough to sleep a parent and a child fairly comfortably.
I've never ridden in a Family Room. From what I've seen of it, though, it may be a bit on the cramped side for the 5 of you, particularly with a lack of floor space for baby stuff. Two of the beds, of course, are short--fine for small kids, but nothing extra, so you'll have to figure out how to manage the baby.
I'm happy to discuss our trip last summer at length (and let me just say, we had a FANTASTIC time), but as to your basic question--how will a 2 year old do? Well, the Empire Builder carries a fair number of families on vacation, so I wouldn't think you'd get a "cold shoulder" for having kids onboard. It's a nice LONG train, so you have a few extra cars to stretch out in--I'm sure the 2 year old will want to walk the entire length (or as much as you can access) several times every hour or so! As with many (most? all?) western long distance trains, you've got the Sightseer Lounge, of which you can make EXCELLENT use--a couple of times we moved from our bedrooms down to the lounge and just camped out there. As you can see from a floorplan (check the Amtrak site), the seats are arranged in semi-circles and such, so you can create sort of an enclosed area in which to corral the kids. The middle day of the trip will have Park Rangers on board to talk and some show-and-tell stuff for kids, but yours will probably be too young for that. But we carried lots of stuff for the kids to do onboard, and our kids were of the ages (including the 3 year old) that being able to sit and do stuff with them (as opposed to, say, dad being up in the driver's seat of the minivan not wanting to have anything more to do with the kids than getting them to sit down and shut up) turned out to be a LOT of fun.
In general, it'll have to be a gut call on your part. I think it's doable, in that there's lots of space onboard to keep you from getting cramped, but the actual logistics (including the tight space in the bedroom itself) may very well make it not worth it.
A final thought: I might lean away from doing it this summer if it's the first train trip for everyone, particularly if it's the first train trip of ANY kind. My wife and I had both traveled in sleepers previously, plus we did a "warm-up" trip (long weekend, overnight on the train each way) to see how the kids did.
Let me know if you have other questions.