10 days in Germany and Austria - thoughts

1) Munich first over Black Forest or Baden-Baden any day. Munich is a nice city, Black Forest is very nice wooded hills. Both deserve 2+ days, but they are opposites. Your choice. Baden-Baden is worth a few hours only if you have nothing better to do.
4) Probably. Ask the car hire company.
5) Berlin is an outlier. Perhaps fly into Berlin, Berlin --> Rothenburg Ob Der Tauber -> Munich etc. and fly back from Vienna. Or perhaps (if you have time) do Vienna --> Prague --> Berlin.

Never spend one night anywhere, 2 nights minimum.
Make sure all your hotels have got parking, Or consider the whole thing by train.

Edit, there is something screwy about the timestamps. The original post is showing as "05/12/16 04:06 AM", but that is tomorrow, my reply is showing as "05/11/16 11:15 PM"!

You have quite a few one night stays right after you arrive. I would try to rework that, if possible. I usually stay my first day in Heidelberg (delightful city)... I pick up my car and go to hotel and spend that first day there. If I am delayed I am not planning a big drive the first day. Then you can leave for Romantic Road next morning. Rothenburg is about 2.5 hours vs 1 hour, but I always want to take my time along that drive and make a number of stops.
I think Baden Baden has little to offer... so many other far more interesting places. It would be expensive to drop the car in Vienna. Just go to Priceline and set up your scenario and you will get price quotes.

If this is your first trip to Germany I think you are underestimating all of the amazing little towns there are to visit in southern Germany. Your trip sounds more like a tour that would be set up to see some highlights, but with your own car you can do a far more interesting trip.

Two nights in Berlin really only leaves you one day...Many here will suggest not doing the Romantic Road, and while I love road trips, not sure this is the best use of your limited time. Staying only one night in places really leaves little time for sightseeing... you drive, check in to hotel, have the afternoon or evening to hang out, then get up the next morning and take off again.

You might consider dropping Vienna this trip. Check out how many miles you are planning on driving. I know lots of places look close, but there are always charming places to stop along the way to enjoy the view or a roadside picnic, etc.

Time stamp shows correct here....

I agree with Terry Kathryn, I think you have signficantly underestimated the drive times. You are trying to cover far too much in too little time.

Right off the bat, consider the effects of jet lag. You are renting a car and immediately driving jet-lagged in (if I can make an assumption) a country you have never visited. You immediately head southeast on A3, which is one of the most heavily traveled roads on the Autobahn network. A few years ago on Main Tower (a Frankfurt news program), there was a feature on the problem of travelers from overseas (ie, Americans and Canadians) frequently getting into accidents under exactly the plan that you propose. Give this some serious thought.

Rothenburg odT, as pleasent as it might be, has been blown out of proportion by the travel industry as something far more unique than it is. Well-preserved historical towns in Germany are not rare at all, and many even maintain all or part of their old defensive walls. If you absolutely must start driving on the first day, consider then at least staying in Büdingen, a lovely old walled town just north of Frankfurt, and work off the jet lag there.

Others have pointed out, the Black Forest is completely out of the way from your general direction of travel. I would check out Baden-Baden only if driving by anyway.

Driving and parking are very difficult in Munich. Don't take a car into the city unless it can't be avoided or your hotel can guarantee parking (and only the most expensive hotels can... for a substantial fee).

Dropping off a car in a different country will net you a substantial fee. And really, I see little advantage in driving for any of your proposed itinerary. With the exception of Rothenburg, which lies at the end of a tertiary rail line, all of your stops are well linked by trains.

Only one full day for Vienna is barely worth the trouble of getting there.

Way too many places. One night =0 full days. Salzburg for 2 nights =1 full day, which you may spend in Hallstatt, not Salzburg. You're going to need a vacation after doing this itinerary.