German Train Travel & Car Rental

I will leave it to others to answer your ticketing questions. Myself I would look it up on the VRM and MRB websites, but the info in English may be limited.
You can look up train times on www.bahn.de

As regards the final bit of the trip, dropping the car off in Lörrach is logical. It avoids any foreign drop off fees which you would get dropping off at Basel airport. Lörrach is a suburb of Basel, it just happens to be on the other side of the border.

Do not stay at a Basel airport hotel. There are no hotels at Basel airport. There are some hotels in France near the airport, and one that calls itself the Basel airport hotel but is actually halfway between the airport and Basel city centre. All are totally inconvenient for Basel city centre. Find a hotel in Basel city centre.
You can see all the hotels on the Basel city website: https://www.basel.com/en/Staying-in-Basel
Expect this to cost s little bit more than hotels in Germany. If you stay at a hotel in Basel city you get free public transport for the duration of your stay, including free transport to the airport when you leave (bus every 7½ minutes).
The train from Lörrach to Basel is easy. It is an S-Bahn (local commuter train), 2 trains per hour, and there are multiple stops in Lörrach. Hopefully one is near your car drop-off.
If you have any more questions about Basel. just ask or send me a message.

Thank you, Chris F, especially for the hotel information. We haven't booked our Basel hotel, yet, until we were sure about the car drop-off. I see Europcar rental return is on Schwarzwaldstrasse 53. These names are going to be a killer for me :) I also see not far from the drop off is an "S" stop "Lörrach Schwarzwaldstrasse" that will take us into Basel.

MRB (MIttelRheinBahn) is a privatized rail company running regional milkrun trains (RB) on the left bank of the Rhein between Mainz and Koblenz (also to Köln).

VRM is a transit district (Verkehrsverbund-Rhein-Mose) which operates the trains in an area from Oberwesel to the Rheinland-Pfalz/Nordrhein-Westfalen border just down river (NW) at Rolandseck, and up (south) the Mosel past Bullay. The VRM website has a pretty good English side.

The VRM does NOT include transport between Oberwesel and Mainz. So their ticket will not work there. However, a Rheinland-Pfalz-Ticket (29€ for 2) will cover all you travel from Mainz to Bonn and up the Mosel to France. Transit from
Rolandseck or Bonn will be by tickets from VRS (Verkehrverbund-Rhein-Sieg).

If you want the minimum connection, There is one at 9:34 am with a change in Cologne. Its not the fastest connection, although the 1st train is an ICE, the connecting train is an MRB that makes all the stops down to Boppard.
If you plan to stop in Strasbourg, why not take the train there, and rent a car when you leave, then drop the car at Basel airport. Again there is a single connection route with an RE train to Frankfurt, then a TGV to Strasbourg.

Thank you for the train advice. I really have tried doing my homework to check it all out but, with all the different passes it gets confusing and some I can only find a German site. I do need a 3 day pass for the Rhine/Mosel area and I thought the VRM was the answer. I read on the VRM site that

"from tariff band 8, the 3-day ticket costs the same as the network ticket. So you can travel by bus and train throughout the entire
network for just 37.60 Euro".

It seems like the Rheinland-Pfalz pass, a one day pass for just my husband and myself, would be needed for Boppard to Mainz. The VRM pass for the rest of the days on the Mosel & Rhine. It doesn't look as though Cologne from Boppard for a day is covered by either of these passes.

Then I also see information about DB Sparpreis pass, and read something about a Lander pass, which could be just another name for the passes I am already looking at. I think I need more reading to figure out the best, most economical options.

Rheinland-Pfalz ticket is a laender ticket, just the particular one for the German land (equivalent of a US state) of Rheinland-Pfalz.
Another option is the VRM daily minigruppenkarte for 22.10 EUR. That covers both of you, but limits you to after 9 am on weekdays.
Going down to Bingen, don't use the VRM pass for the day, just 2 one-way tickets is only 20.40 EUR.
Going to Cologne for a day trip, best would be no VRM pass for the day and travel with a Quer-Durchs-Land ticket of 52 EUR. Covers both of you after 9 am weekdays, all day weekends. To get a slightly earlier start on weekdays, buy two 1-way tickets to Koblenz for about 12 EUR total.

The Bahns in germany can be brutally confusing, especially for visitors who are unfamiliar with the different regional trains, the direction/geography of cities, and the ticket pricing system. I have had to pay fines many a time for getting on the wrong train.

However, I still think it is better than renting a car for the cologne and rhine region. The local rhine/mosel local trains were totally convenient, it isn't difficult at all going one way with the train and back by boat or bike. getting around that area is really easy.

Also, check out flixbus. they have a bus from cologne to strasbourg daily for 18 euros. It takes aout 4:45 mins, which is a bit longer than the train or diving but immensely cheaper.

coming back I took a bus from strasbourg to mannenheim, then a train from there to mainz. I would see if that option is in reverse.

I wouldn't want to have to deal with a car in the cities of cologne or strasbourg.

I wish the trains in germany weren't so expensive.... ::sigh::

Thank you, I will look into these other options. No worries about leaving before 9 a.m... anywhere. On vacation, you know. The train system is beginning to make more sense when I think of the laender ticket, Rheinland-Pfalz, as "statewide." It's particularly confusing when there are local & regional, "statewide" and countrywide trains and they all also operate regional trains. I will also look into a "minigruppenkarte" (one-day in valid area) and Quer-Durchs-Land ticket, now that's a new one I haven't heard of. (One-day ticket on Bahn regional trains only, but throughout Germany), I just have to figure which and where we need i.e. regional trains ICE, or IC. The fastest, with the least amount of connections works for me.

June, you wrote "and some I can only find a German site". Maybe you didn't realise, but many of these sites are multilingual, there is rarely a separate English-language site. Look for a small menu at the top listing languages, or a list of abbreviations to click on ("de fr en" for example), or a Union Jack flag to click on for English. Possibly not everything on the site will be in English, but a lot of the useful information will be.
The DB site ( www.bahn.de ) and the VRM site ( www.vrminfo.de/ ) are examples of this.

DB = Deutsche Bahn = German Railways = the national railway company.
VRM = Verkehrsverbund Rhein-Mosel = Rhine-Mosel Transport Association. A "Verkehrsverbund" is a fares pool where all the transport in and around a city (bus, métro (U-Bahn), trains etc.) all take the same tickets, and you can change from (for example) bus to train without getting a new ticket. Within a Verkehrsverbund their tickets apply on the trains, not DB tickets. If you are on a train that goes outside the Verkehrsverbund, you need a DB ticket, or a Länder (state-wide) ticket.

A map of he VRM rail network is here: http://www.vrminfo.de/fileadmin/data/pdf/2016/Schienenliniennetzplan_2016.pdf
A map of the Nordrhein-Westfallen rail network (i.e. where you can use a NRW Länder ticket) is here: https://www.bahn.de/p/view/mdb/pv/deutschland_erleben/nrw/regionales/mdb_111068_regionalverkehrsplan_nrw_2013.pdf