Rhine/Black Forest/Cuckoo clock

Day 1: Cologne. My favorite German City, although I could see Berlin overtaking it. Stop at Papa Joes Biersalon in the Altermarkt, and take a stroll over the railway bridge (Hohenzollernbrucke) next to the cathedral.

Days 3-5. Not Koblenz. Boppard or St Goar. Take a Rhine cruise on the K-D line. You can go the slow way up river to Bingen, then buy cheap return ticket to Oberwesel where your VRM ticket becomes valid. Easy to get around on trains and buses with a VRM 3-day Tageskarte for 37.60 pp. No problem getting to Burg Eltz. Take the train to Koblenz and change there for the Mosel valley rail line. If a weekend, there is a bus from Hatzenport station, otherwise hike (on a nice day) from Moselkern station.
Also on weekends check out wine festivals in the Mosel valley with the event finder at http://www.mosellandtouristik.de/en/

Days 6-10. Just make sure where ever you book that they offer the KONUS card which gives you free bus and train transport in the Black Forest region. Triberg is cuckoo-clock central. Triberg to Berlin is 7 hours on the train with a single change in Offenburg if you take the 10:14 regional train from Triberg.

Your'e doing quite well on your own so far!

Cologne is not unreasonable (nor is it a must see) but the Ruhr cities to the north are very industrial, so probably not.

Koblenz, is a good location, maybe a little short of hotels in the centre. The place I used was a few minutes on the bus but turned out to be used for army officers lodging during the week and Russian tourists at weekend (has the german security service thought this out?). Personally since it's only an extra 70 minutes from Cologne I'd go straight there on day 1. I'd also look at the district of http://www.koblenz-guels.de/ as base. This is a village on the Mosel on the edge of the city, it has a good bus and train service into the city. Once there you can't miss a rail bridge over the river, there is actually a public footpath bolted onto one side and you can walk over, there are even more buses to the centre at the other side. Hopefully you've already worked out that all local transport authorities and states have their day tickets (usually after 9:00) which get cheaper as more people join the group. In this case the locals are VRM http://www.vrminfo.de/en/ and the state is Rheinland-Pfalz

https://www.bahn.de/p/view/mdb/bahnintern/angebotsberatung/regio/laender-tickets/pdfs/mdb_209917_basisflyer_rlp_2015_dina5.pdf

all the places you mention for short visits are fine (some may be just an hour but with a transport day ticket you can bounce around as you feel the need).

Second phase, to Freiburg by all means, stay there no. You heard correct about free transport in the Black Forest region but Freiburg doesn't give it (although you can use the pass on their transport to visit). http://www.blackforest-tourism.com/info/KONUS you need to use the german link to check whether a place issues it (simple list at bottom of the german page). The bigger working towns tend not to be members. Back to Koblenz, there are lots of options to get to Freiburg but one stands out, train EC9 departing Koblenz at 11:48 arriving Freiburg at 15:01. This has a Swiss glass roofed panorama car in 1st class. The bahn.de website is currently showing 1st class prices from €39,90 (it is you farewell tour....) but that doesn't let you specify the car, go to a DB Travel Centre in Berlin and book there, ask for seats in car 264. If they can't do it just take any 1st class reservation on the train and change seats on the day.

Place to stay, there are Konus towns and villages along the Rhein railway but I stayed higher up on the line which heads east. Try Titisee, trains every 30 minutes from Freiburg. Lakeside and the altidude means it can beat early morning autumnal fog which places lower down may have. I stayed here some years ago, the food was good, they have an indoor pool and are across the road from the station - and they usually have a range of multi-night special package deals which might be good value if you can make them fit your plans http://www.parkhotel-waldeck.de/en

Konus puts Triberg, Hausach, Schweningen and much, much scenery in easy trips from Titisee, Freudenstadt has a nice centre (and a very nice brewery pub restaurant) but only slow local trains so takes a while. Don't forget at least an afternoon in Freiburg (or late afternoon & dinner). Konus is also valid into Basel by train, I went from Titisee over Feldburg mountain by bus and caught the Basel train at Zell. You can also use regional trains (at least hourly) via Freiburg. Note Konus is not valid on IC/EC or ICE.

Journey back? A full day by train. Or easyjet flight from Basel (use the special bus from Freiburg Hbf to the airport).
Or night train Freiburg- Berlin. NightJet departs 21:58, arrives a little after 6:00 next morning. http://www.oebb.at/en/angebote-ermaessigungen/nightjet

"Days 3, 4 and 5: Stay in the region, visit Marksburg and Rheinfels, or others. I’ve also seen the other fork of the river recommended, with Burg Eltz and Cochem. I suppose we will have to make a choice here."

That other "fork" is the Mosel River.
http://www.mosel-reisefuehrer.de/17-highlights/556-moselhighlights-englisch

Besides Cochem and Eltz, Trier is a great destination as well. So is Bernkastel.

The Rhine towns have local wine festivals every weekend in this town or that. Great fun.

I would suggest you allocate maybe 4-6 days in this area, 2-3 on each river. If you like biking and hiking, maybe more.

You do not need to visit the Black Forest to shop for cuckoo clocks. St. Goar has a shop.
http://www.bugbitten.com/images/40d1dd91595f074c0cbda6b4c3a0ace5/Contiki_Amdam_to_Venice-117683/Contiki_Amdam_to_Venice-4082688.jpeg

I like the Black Forest but you should know that there are lots of interesting places you might visit in between Berlin and the Rhine Valley, places with nice scenery, old-world architecture, half-timbered buildings, etc. See some of these very fine places instead on your way to/from the Rhine/Mosel, and you avoid those extra-long train journeys you must make for the Black Forest.

Hannoversch Münden
Goslar
Büdingen
Hameln

German Half-timbered-House Road towns (most are accessible by train.)
Fairy Tale Road towns (also many accessible by train.)

We were very underwhelmed by Cologne, you could make it a quick stop-over to see the cathedral and head on south. The train station is in the shadow of the cathedral so easy in and out. In St Goar, I recommend the Rheinhotel with the little balcony room overlooking the river. It's not the plushest of luxury accommodations, but it's a million dollar view. Easy train transportation between towns in the region, or rent a car and explore at your leisure (I wish we had).

Thank you Sam, SteveB, Russ and celeste for the help. Particularly all the detail you've included, really is much appreciated.