Help with Itinerary Please....

I like your original plan best. Love the Mosel valley area. Might even think about about spending more time there, perhaps dropping 1 day in Paris. Reason is it is good to be there on a weekend where you can find a local festival, there is always one somewhere on the weekend. And biking along the Mosel is great. This site will give you some ideas. It has a search function for events on days when you are in the area.
https://www.mosellandtouristik.de/en/
With a car, you can also change things on the fly.

Thank you, Sam. Your advice and the link are really helpful. I don't really want to shorten Paris by a day. The day before we leave Paris is my birthday and I'd rather spend my birthday in Paris than spend it in transit. If I can make it to the Mosel area by noon, do you think the weekend festivals will still be going or will they be winding down?

In Traben -Trarbach there's an "Altstadtfest" that weekend (June 15-17) - their "old-town festival"; street music, food, wine. Saturdays are typically the most lively day for wine fests.

If you follow the river all the way from Trier to Koblenz and then to Boppard you will be on the river for a very long time. Better check that out closely. The drive + Beilstein + Cochem + Burg Eltz on one day... overly optimisitic? Far too perfunctory? Both?

"I'm worried that little German villages will be a bit of a let-down for the teens."

Take them to a wine garden in Boppard. They can't do that at home.
https://media-cdn.holidaycheck.com/w_1280,h_720,c_fill,q_80/ugc/images/96598b3f-4507-382e-b101-61df5a9a79a1

Do the Klettersteig just outside of Boppard.
https://www.tripadvisor.com/Attraction_Review-g198627-d1812925-Reviews-Mittelrhein_Klettersteig-Boppard_Rhineland_Palatinate.html

Go to the Burg Guttenberg falconry show - the castle isn't far from Heidelberg. See photos here:
http://www.bensbauernhof.com/germany2012.html

Boppard's Chairlift offers a small rush. Great views.
https://www.tripadvisor.com/Attraction_Review-g198627-d3306006-Reviews-Sesselbahn_in_Boppard-Boppard_Rhineland_Palatinate.html

Gedeonseck terrace near the top: http://ext.pimg.tw/samlucky5711/1379089584-1923285561.jpg?v=1379089585

Trier is a great place to visit if you and the kids are into Roman history, but if not, I'd give it a skip. Spend any time you save on the Rhine/Mosel region.. The Rhine hiking is great - try the Rhine Castle trail:

http://www.romantic-germany.info/RheinBurgenWeg.6751.0.html

Russ, thank you so much for such a thoughtful reply. I really appreciated the suggestions and the links you included. The descriptions and photos made me really excited for the trip. I don't know that either the kids or I are into Roman history. My German teacher in high school came from there. When she taught German art and architecture, she would include the Roman architecture from Trier. It seemed so different and far away to that young girl from an off-the-grid part of far, far Northern California. I think you're right that we should probably stay the whole time in Boppard. Any thoughts on how best to get there from the west?

we should probably stay the whole time in Boppard. Any thoughts on how
best to get there from the west?

Train from Paris Est June 17: 9:06
Arrive Boppard Hbf 13:38
(One change of train in Kaiserslautern)
Saver Fare is currently €50 from DB

https://reiseauskunft.bahn.de//bin/query.exe/en

With 4 nights you have a nice choice of outings from Boppard.

MOSEL outing:

VRM is the local transport authority and sells most of the local tickets you need.
A day trip up the Mosel by train from Boppard to Cochem and Moselkern (station for the walk to Burg Eltz) would cost €23 round trip from Boppard (VRM mini-group ticket/day pass, buy at Boppard station ticket machine.)

RHINE outing, east bank:

Use the Boppard ferry crossing and walk into Filsen so you can use the east bank train line to Marksburg (7-minute train ride.) The VRM mini-group ticket price for this short trip (price level 3) is €15.30 for the round trip from Boppard including the ferry. Buy the ticket at Boppard station before you board the ferry; trip starts from "Rheinfähre Boppard" (ferry dock on boppard side.)

RHINE outing south to St. Goar (Rheinfels Castle) + Oberwesel (cool walk around old town wall):
You pay €19.70 (price-level 4) for the VRM mini-group ticket.

RHINE outing to the north: Remagen + WW II museum at Remagen Bridge - VRM mini-group Ticket, €23.

Outings to Mainz, Bacharach and Bingen to the south are outside the VRM zone; the Rheinland-Pfalz ticket covers your travel day for €24 + €5/additional adult. Buy it at the station too.

Both VRM and R-P ticket day passes are valid on local/regional trains and buses (and some ferries) and have an after-9 am travel rule on weekdays.

Train service is hourly or better in each direction no matter where you are on the Rhine and Mosel.

You can pick up a car in Koblenz (north of Boppard) or Emmelshausen (small town west of Boppard - take train up the steepest railway north of the Alps from Boppard station to get there - kind of a cool little train trip) if you like for all or part of your Boppard stay - or when you are ready to leave on Day 12. Of course it's quite easy to get a train to Heidelberg as well.

Omg, Russ. Your information is so wonderful. I'm really grateful. Are you thinking its better to go carless along the Rhine and then pick up a car when we're ready to leave the area?

"Are you thinking its better to go carless along the Rhine..."

That's a choice that depends partly on your preferences. The Rhine/Mosel region (like many parts of Germany) is especially well connected by rail. Almost every little riverside hamlet has a station. Here's a local train map (zoomable) for your reference. Train service is frequent and dependable. The train information I provided for day trips out of Boppard is there to inform your choice. Besides preference, the location of your room(s) or apartment is a factor... for train users, digs that are too far from a train station will make for long pre- and post-outing walks or additional bus trips, so location needs to be a consideration. By train you'll have more walking to do on your outings, but in my eyes that's a good thing in this area - the towns and the scenery are more enchanting on foot. (BTW, hiking is really good on the Rhine - and the weather perhaps more dependable than in the Alps.) If you'll be doing any wine tasting or enjoying a beer or two while you're "unterwegs", the train is really good insurance against accidents or a DUI case.

"...and then pick up a car when we're ready to leave the area?"

Maybe, maybe not. I've done both car and train but have developed a strong preference for train travel. I don't like driving or riding in a car over long distances over several days (like the Boppard - Heidelberg - Rothenburg - Romantic Road - Munich - Alps route you have in mind) as it's just a lot of work and unnecessary responsibility, in my eyes - with lots of variables that can jeopardize one's plans. I don't find driving or parking in any town the size of Heidelberg or larger enjoyable whatsoever. And I empathize with the back-seat dwellers who I've toted around on such trips as well (even my in-laws, heh.)

"We could... go south to Tubingen, around the Schwabishe Alb, etc. to Neuschwanstein and then to Munich. Ugh."

I know you meant the decision - but the drive itself makes me go "ugh." I'd much prefer the train for trips like this - I figure, why not just enjoy the asphalt-free scenery, read a book, break out a picnic on the train? With some planning, most of the Romantic Road towns can be visited by train as well, and stopovers for a couple of hours here or there can be worked out too.

Russ, I just realized you are from Paradise! My grandma was from Chico (Richvale as a kid). I spent a lot of summer vacations there and she always used to take me for a ride up to Paradise. Beautiful town.