Can you "check" a carseat like you can on a plane? --depends on the stations you're traveling between, some have checked baggage service, for other trains you'll end up putting your baggage into luggage areas on the train. If the latter, you'll have access to your bags the whole trip. Red caps for Amtrak are phenomenal and especially helpful if you have a baby strapped to you.
Do you keep your bags with you all the time or are they stowed somewhere?--see above. There's also a huge overhead shelf and lots of foot room. You can tuck a stuffed backpack under your foot rest easily or have it past the foot rest on the floor.
Meals?--included with the sleeper cars, available for purchase otherwise, they run about 20% more than comparable meals elsewhere. You can also pack food, especially if you'll be doing a shorter trip (i.e. 13-24 hours)
When they make stops, how long are they?--depends, since they don't allow smoking on the train, with longer trips especially they'll try to include regular longer (15-30 minute) stops. Other stops will only be as long as they need to get passengers on and off.
How noisy are the cars?--a constant low thrum, quite soothing
What is the seating like?--roomy. I don't know about people who are morbidly obese, but my mom's a 3W and she fit just fine. For sleeping in the coach chairs, you'd want to put a bag on the floor past your leg rest in order to get maximum stretching out comfort.
Do they require car seats for kids? No seatbelts.
They do require that you stay with your LOs at all times. My toddler had an easier time walking on the faster trains than any of the adults except the conductors.
Do they have outlets for charging phones or playing movies on my laptop if I'm bored silly with holding a napping baby? Most trains do have two outlets at each seat, some still only have a few outlets spaced throughout the train for vacuuming. You probably won't have consistent cell coverage though. If you've got a LO who hearts outlets, you might want to be ready to put something boring over them.