Rothenberg ob der Tauber and Nuremberg

Berg and Burg are not interchangeable (nor pronounced the same). In German it's Nuernberg (Nürnberg) and Rothenburg. (The -mb- comes because when Germans pronounce the -nb- sound, it sounds to Americans a little like -mb-, but not really.

You don't need to go back to Nürnberg to go to Stuttgart. Take the shuttle out to Steinach and catch a train from coming from Würzburg to Ansbach. From Ansbach there are multiple train connections, both long distance (IC) and regional (RE), directly to Stuttgart.

I've changed trains in Steinach to Rothenburg three times. I've never had problems with the connection times. Before you get to Steinach, get up, collect your luggage, and go to the door of the train ready to get off when the train stops. Coming from Rothenburg, you'll arrive at the farthest out platform (5) from the station and have to go through the tunnel to the next platform (3/4) for the southbound trains. Just follow the herd from the train to the tunnel.

I always opt for trains in Germany vs buses when I have the option, but if you are concerned about the change in Steinach, the bus goes directly to Dombühl, where you can catch the same train to Stuttgart that you would have caught in Ansbach.

We are a little confused by the travel and ticketing options (trains)
(trains/buses) between the two cities, eg., multiple connection with
short time intervals,

The VGN Tagesticket Plus is a day pass good at any hour for travel all day long within the VGN zone - and Nuremberg > Rothenburg lies within that zone; price is €23.90/2 adults:

https://www.vgn.de/en/tickets/all-day-ticket-plus/

The DB schedule provides short layover times between trains for this trip. I've done it several times and never missed a connection. The final train to Rothenburg from Steinach will wait for passengers arriving in Steinach on their way to Rothenburg.

and understand we will have to return to Nuremberg from Rothenberg after a one-night stay to then travel onward to Stuttgart.

This is untrue. You will normally have changes in Steinach and Ansbach on the way to Stuttgart. Alternatively there is a bus to Dombühl with a direct connection from there to Stuttgart. The DB site will provide these schedules.

I don't have much of a comment about sleeping in Rothenburg. Sleeping is sleeping... and you are leaving the next morning for Stuttgart. I don't see a problem with this plan... smarter than returning to Nuremberg to sleep, I suppose.

Well, it's definitely nice to stay in Rothenburg overnight, but you're right in that it would involve a bit of logistical planning. Have you considered renting a car for a day or so, so that you could pick up the car in Nuremberg, drive to Rothenburg, spend the night, then drive back to Nuremberg, drop off the car and proceed on your way.

it shouldn't cost too much and would definitely make things easier.

Thank you Lee and Russ. I will review the map and travel times to process what you have said. The takeaway is we do not have to return all the way back to Nuremberg.
It seems we would only buy a single VGN ticket the morning we travel to Rothenburg (360 minutes) and different ticket(s) traveling to Stuttgart. Are the buses accomodating to traveling with luggage?

an entire evening in Rothenberg instead of an extra night in Nuremberg
is worthwhile

Is catching a train at 11:05 PM an entire evening? You would be back in Nuremberg by 11:30 PM.

Or, leave your bags overnight in Nurenberg https://www.bahnhof.de/en/nuernberg-hbf/locker and just take a small handbag for an overnight in Rothenburg.

Also, since I don't know where you are getting your train info, here is the link to DB site https://int.bahn.de/en/

It seems we would only buy a single VGN ticket the morning we travel to Rothenburg (360 minutes) and different ticket(s) traveling to Stuttgart.

I do not understand "360 minutes." These morning departures on Dec. 3 from Nuremberg to Rothenburg are samples taken from the DB site:

8:37 - 9:51 (two changes of train, 74 minutes)
9:21 - 10:51 (two changes of train, 90 minutes)

The VGN ticket can be bought from a ticket machine at Nuremberg station or using the app. It is available for shorter trips at lower prices... but you should buy it at the maximum price (€23.90) for this relatively longer journey. Do not pre-purchase this ticket far in advance... the day before for the travel date, or the day of travel, will be just fine. It is non-refundable if you change your plans.

Getting to Stuttgart: I am partial to train travel generally and also in your case, partly because you may or may not find it easy to locate the right bus stop, then get off where you switch to the train, wait for your bags to be outloaded from the lower storage compartment (if the bus driver requires that) then tote them into the station. It's just simpler to get off and move to the correct train platform. There are 8:05 and 10:05 departures from Rothenburg which take 3 hours to reach Stuttgart by regional train. 2 changes are required. Both options have a short layover in Steinach (the station where you changed trains the day before just before to board your final train to Rothenburg) and a long layover in Ansbach. I would use one of those.

There is also a 9:50 departure, but this one takes 3 hrs 48 minutes because it routes you via Würzburg rather than Ansbach.

If you can leave after 9 am, your regional train ticket should be the "Day Ticket for Germany" (€55/2 adults.) This is a day pass that covers any regional trains you like anywhere in Germany for one day. It is not available for trips before 9 am on weekdays. You are not required to keep a schedule as it is completely flexible for any regional trains you wish to use on this day.

https://int.bahn.de/en/offers/regional/day-ticket-germany

Buy it at the ticket machine at Rothenburg station or use the app. This ticket is also non-refundable, so purchas just before you use it or the day before for your travel date.

To see all the train journeys I mentioned above for Rothenburg > Stuttgart at the DB site...

1) click on "local transport only" at the mode of transport box. Then...
2) click on "user defined" and deselect the buses.

If you take the Rothenburg to Dombühl bus and have a lot of luggage, it will probably have to go in the storage area under the bus. (I've never worried about that. I always bring my less-than-regulation-sized carry-on backpack onto the bus with me. There is sometimes some overhead space or often empty seats; at worst, my backpack can sit on my lap.)

Here is the map of the station in Dombühl.

The bus from Rothenburg arrives at Bushaltstelle 2 (bus bay 2), beside the road into the station. It's about a 50m (50 yd) walk to the front of the station and another 50m through the tunnel to platform 4, where, according to the Bahn website, you will board the train to Stuttgart. According to the schedule you will have 15 minutes to make the 2 minute walk.

Of course, the bus could be late arriving so you might not have all 15 minutes, but I was once on a bus to Bad Tölz, and we were delayed in route when a passenger came on-board and spent 10 minutes looking for her pass. We arrived at the station just as the train was supposed to leave, but the driver had radioed ahead and they held the train for a few minutes for us to board.

I was looking at the VGN tickets that appeared to be good for 360 minutes immediately after purchase which of course would need to be selected at the station just before departure. I understood this was the type of ticket to buy for Nuremberg-Rothenburg train.

Aha. That is a standard one-way transit-authority ticket valid for one person on one route. Such tickets expire after a certain period of time. The Tagesticket Plus I suggested is a network ticket valid for 2 persons, not restricted in terms of where you travel as long as you stay within the VGN zone; it replaces that standard one-way ticket and is valid for the entire day.

TYPO ALERT:

I wrote that "There is also a 9:50 departure..." from Rothenburg to Stuttgart. The correct time is 9:05.

So you suggest the Tages Ticket Plus covering both of us , one-way and to buy it that morning at the train station. We would not be traveling any farther than Rothenburg so the 360 minutes would be plenty of time, correct?

You need to forget about the 360 minutes. The Tagesticket Plus is not a normal ticket. It is a DAY PASS, and the 360 minute cap for use, which applies to normal tickets, does NOT apply. I have copied and pasted the wording below from the webpage I linked to above at 8:39 am. (You can click on that web page for additional information):
 
Period of validity: one day or weekend (until last bus/train)

Two one-way tickets for the Nuremberg > Rothenburg journey (price level 10) would cost you €26.84. And you could buy those tickets instead, but it would make no sense to pay that much, which is why I recommended the DAY PASS for two priced at €23.90.

Thank you for all your great help Russ! I am copying and pasting your suggestions to a note I have for Rothenburg.