Hi, and welcome to the Forum!
I'm sure you will enjoy this foray into a modernized version of an old transport mode. I do like to set realistic expectations - maybe even lower than realistic - so that perhaps you will be prepared for the worst, hope for the best and be ready for an adventure that can almost always be spun into a positive experience.
Your choice for the family bedroom is excellent. The child bunks are small - but it sounds like you will all fit. It will be snug, but the room is pretty large. The best thing about the Family Bedroom is that you have the full width of the train car to enjoy BOTH sides of scenery from. The down side is that it's on the bottom level of the Superliner trainset, so you won't have the advantage of a higher-elevation view point. In fact, the lower level of a double decker like you'll be in sits slightly lower than most single level trains. But, don't fret. You'll be near a Sightseeing Lounge where you will have access to panoramic views of your journey. The other advantage of being on the lower level is that you'll be close to your baggage racks, located near the entry door to the train. You'll also be close to restrooms and showers that fewer folks will bother walking all the way downstairs to use.
As for packing, I think travel size everything will be appropriate. Even though you'll have pillows and blankets, a supplementary set from home can always be helpful. Amtrak sells a "comfort kit" that includes an airline-style blanket, air pillow, eye shades, and ear plugs for around $8 per kit. It's a nice little kit, but you don't
need it in a sleeper. You will need to bring your own toiletries.
Folks also suggest bringing a little duct tape. It may come in handy to stop things from clanking or squeaking in the room.
As far as which car to be in, you may want to be in the car closest to the diner, just for convenience sake. Personally, I like taking a walk through all the cars, but with kids in tow, may not be as much fun for you as it is for them.
So, now, my low expectations. Expect to miss a connection, though you only have one in Chicago. Expect it so you aren't critically late to your destination (like by a day). They will take care of you, but patience is a must. The Empire Builder has also been chronically late in itself with all the oil boom in the Dakotas taking up so much rail space. Believe it or not, the companies that own the tracks want to run their own trains on them. There is also the prospects of mudslides from the Spring Thaw that can slow you down. In all this, it is much better that you are traveling West rather than East.
If there is such a diversion, be prepared for a bustitution. Again, they will help you with every step, but communication may be poor, so be vigilant.
Now that I have scared you into wondering if you should even go or not, I KNOW that it SHOULD be better than that and you will have an incredible journey. It's the unexpected that makes the rail journey memorable. It's the people you'll meet that will make it fun. So, sit back, relax, and enjoy the ride!