Cologne Visit Sunday/Monday

I make it a point to get to Cologne every chance I get, so I'd say Go For It! It's a 1h - 2h30m trip on a Sunday, depending upon which train you take, then from 2h15m - 3h15m to get to Stuttgart on a Monday. I saw a few trains that required reservations, but most didn't; whether or not they're needed I'll leave up to the people that live there. Worst case - you'll be standing for a few hours - a chance to stretch your legs after that plane ride ;-) I'd try to get a reservation for the Stuttgart trip while in Frankfurt, since that's a longer trip. Reservations are cheap enough to me.

ICE trains require reservations, but the act of buying a ticket gets you a reservation and seat. There are a handful of nonstops get you there in 49 minutes, and a ton of trains with 54 to 67 minutes with 1 or 2 stops. The std price is 66 Euro, but if you want to roll the dice, there is a 10:32 with a 35 Euro nonrefundable fare.

There is an Ibis hotel as part of the Cologne train station which is next to the cathedral.

"ICE trains require reservations, but the act of buying a ticket gets you a reservation and seat." That's not true in Germany, unless you're riding on Thalys or certain specialty trains. The normal ICE between Frankfurt Flughafen and Köln does not require reservations. If you want one, you have to buy a separate reservation, which only costs a few euros. I don't see any major problems with the plan. The only thing I would add is that if you purchase your tickets in advanced, give yourself a very large window of time after the flight to catch the train. If you don't mind paying the full price, just buy your tickets when you get to the station, and hop on the next train to Köln. It's a very busy rail corridor, so you shouldn't have to wait very long. To save money, I would definately purchase the ticket for Monday in advanced. As noted previously, reservations are not required. However, I recommend buying one. Not because the train will be so packed that you can't find a seat, but because knowing where your empty seat is in advanced will allow you to simply board the train and move straight to it. This is particularly helpful when carrying luggage, and when boarding at the Fernbahnhof (airport long distance rail station), when many people generally board the train simultaneously with lots of luggage.

ICE trains require reservations When did that start? I've never had a reservation for an ICE, maybe lucky but I've never stood on an ICE either. I once stood on a TEE but never on an ICE.

Try to figure out what time masses in the main part of the Dom are on Sunday, since it will affect your ability to visit. I don't know if you can get to the Treasury during a mass, which would be a good use of your time. The Treasury (admission charge) is not to be skipped if the Dom (free) is your main interest. For the tiny amount of time you have in Cologne, it can be considered to be 100% walkable. Make a plan for what you'll do after the museums close for the day-which may suggest walking right by interesting things until that hour. You can barely scratch the surface of Cologne in 24 hours. I personally would check the rates at the Hilton (which is essentially at the Hauptbahnhof, but out of sight of it), but many posters here want the rockbottom price.

"...on a TEE" That's going back quite a bit. I've stood on a ICE, 2nd class, maybe for an hour, depends on your luck then.

Yes, the TEE was a couple of years back - I mean decades - several.