Some advice, hiring a car or public transport

will you be flying or catching a train to leave? From Munich or back through France or up to Frankfurt?

Memmingen is an interesting starting point - you'll be passing near on the way from Munich to some parts of the Black Forest.

In other words, are you questioning using a car in the Black Forest only, or for the broader area?

Using a car or the trains are both valid modes in the Black Forest area, but they will give you different ways of enjoying the area.

If you are staying overnight in many towns (but NOT Freiburg im Breisgau among some others) in Baden-Württemberg you will be given a KONUS card which give you free transport on regional trains and buses in B-W including the trams and buses in and around Freiburg im Breisgau.

I know you are looking for suggestions on what to do - but let me ask what do you like to do? What is attracting you to the Black Forest? Tell me what you imagine and I may be able to help you choose car vs train.

Baden-Württemberg takes up much of Germany's southwest corner:
https://cdn.britannica.com/34/114934-050-A42B2937/Baden-Wurttemberg-locator-map-Germany.jpg

Have a look at also at the railway density within B-W below:
https://karteplan.com/deutschland/land/baden-wurttemberg/schienennetz-karte-von-baden-wurttemberg.jpg

The Black Forest, which makes up most of the SW quadrant of B-W, is well-covered with railways too:
https://prospektbestellung.toubiz.de/media/prospekt/file/5736289_KONUS_GB_2022.pdf

Service is frequent and dependable, though sometimes there are delays. So the short answer is "Yes." You can easily visit the Black Forest across 7 days by train, and you will still have many nice destinations left over for additional train-based visits on future trips.

The problem with suggesting places to visit is that there are way too many candidates competing for your time, and we have no idea what interests you. Also, there are dozens and dozens of nice places in the remainder of Baden-Württemberg that might interest you as much or more than those in the Black Forest.

The 3rd link I provided explains the KONUS program - obviously, the free train travel within the BF is an advantage there.

I suggest you check out a Germany travel guidebook like Fodors or the Rough Guide or whatever you can find at your local library and check into their recommendations for Baden-Württemberg area to see what interests you most. The RS Germany guidebooks don't go very deep into individual German states in the way that the more comprehensive guidebooks do. Once you know where you are going, you can work out the transport details.

Within Munich, especially around Oktoberfest, a rental car is not an advantage, I recommend to take the Flixbus from Memmingen Airport to Munich. Use public transport while there or for day trips.

For the second part I recommend using a rental car for more flexibility - either from leaving Munich on, e. g. from station Munich Pasing, or after return to Germany. Avoid cross-border rentals.

Important:

1) Ensure valid driving permission for all countries you want to drive in. For Germany it is valid driving license and maybe IDP. Check regulation: Validity of foreign driving licenses in the Federal Republic of Germany

2) If drive includes Austria check need for toll sticker / vignette: https://www.asfinag.at/en/toll/vignette/

3) Make yourself familiar with rules and road signs.