Planning Questions: Car Rental In Cologne and Stay in Rhine Valley

No way drive from Cologne airport to the centre of Cologne. Cologne cathedral is directly in front of the station (Köln Hauptbahnhof). It is a pedestrianised area, you can't park anywhere near there.
I can't say for sure, but I would expect any station of the size of Köln Hauptbahnhof to have left luggage lockers.
St Goar and Cochem are close enough together that there is little point in splitting your time between the two. choose one and stay there, and visit the other from the first.

You then come to the big question, whether you need a car. Köln --> St Goar / Cochem --> Trier --> Luxembourg could all be done by train. But, if you want to visit small places outside the Rhine and Mosel valleys a car is useful. And 3+2=5 nights in one place (St Goar or Cochem) may be too much without other places to go to.
My suggestion: hire a car from St Goar / Cochem to visit the area, drop it off in Trier, and do all the other legs by train.

Hi Chris, Thank you for the quick response and excellent suggestions. Sorry, but we are from Southern California and we can't be too far away from a car/driving. It's in our blood and psyche. But will try to wean ourselves since it does make sense to use public transportation when it is convenient. We will rethink our plans for sure. Thanks again. Dee

Trains = no worries about how to pick up, drop off, or park your car. No contracts. No insurance. No big bills for tiny scratches. No parking or traffic issues...

"Sorry, but we are from Southern California and we can't be too far away from a car/driving. It's in our blood and psyche. But will try to wean ourselves since it does make sense to use public transportation when it is convenient."

If you're willing to wean yourselves, you will probably find it very easy. As Rick Steves says, "Trains remain the quintessentially European way to go..." and there's really no better place to give that option a try than Cologne and the Rhine/Mosel valleys.

There's a direct train called the S-Bahn that leaves from the CGN airport station every 30 minutes or so for Cologne's main station. Once there, you store your bags there in a very modern way while you see the cathedral. There are instructions in English. See video. There are ticket info counters with English-speaking staff if you need help. The ticket machines speak English too and can help you with a ticket or just a schedule (which you can print.)

Two can make the trip from CGN to Cologne on the S-Bahn and then from Cologne to St. Goar on one ticket/day pass (around €45 - €55.) There are direct trains from Cologne to St. Goar if you want to keep it simple - the MRB trains. Grab any seats you like on those and enjoy the river scenery on the way south. Signs in each train car announce the next stop (this train will be stopping next in Bacharach.)

Outings from St. Goar to the other Rhine towns (Boppard, Oberwesel, Bacharach, Braubach, and Rüdesheim are probably the most popular) are all very simple as well. All these towns are quite small and the stations are easy to find and reach. Use the St. Goar ferry (which runs all day long) to cross the river and board the train there for the last two towns above. A day pass will cost between €22 and €28 depending on your destinations and will include the ferry.

Getting to Cochem is easy too - just one change of train in Koblenz. Outings from there and transport to Trier and on to Luxembourg are equally straightforward.

http://www.bensbauernhof.com/accommodationsrhinemosel.html is a great source of info for the area--Bavaria Ben.

Depending on your interests, Cologne has a lot more than the Cathedral. That Cathedral is famous for being next door to the train station. As noted, it's not a place you want to take a car. We also rented at the train station for our departure from Cologne. Cologne is a much richer destination than Luxembourg City.

As noted, your Rhine / Mosel stops are fine with a car, although you should not assume that your hotels or B&Bs have free onsite parking. I once parked on the street in Trier, and it was very risky from the point of view of scratch and dent possibilities. You will learn in indoor garages that you want the smallest car that can carry your luggage.

Thank you all for your constructive suggestions. It is friendly people like you folks that help others to make better travel plans. Very much appreciated. We will need to rethink our itinerary and take into consideration all the helpful tips. Thank you so much!