IMO, the fastest and easiest way for the first half of your trip would be something like:
Frankfurt -> Bonn by train. There is a IC/ICE train each two hours from the main terminal in Frankfurt airport to Bonn city-center, 10 minutes walk from Beethoven house. The trip takes ~ 2 hours. If you fly with Lufthansa from US, you can have the train ticket included in the flight price, or at least heavily discounted (when booking, Lufthansa offers you Cologne/Bonn train station as a destination).
I’d guess this is at least as fast as coming from Düsseldorf to Bonn.
The public transportation in Bonn is very good and it is integrated with the one in Cologne (that is, you can use one ticket to go from one to the other via train, tramway or bus). But 3 days for Bonn is too much, unless you want to have a good look at each piece exhibited in each museum in Bonn. So I second the above recommendations: don’t miss Cologne! There are several museums worth visiting (I’ve visited Wallraf museum a couple of weeks ago and I really liked it), not to mention the cathedral.
For sightseeing, there is also Drachenfels, a hill next to Bonn easily reachable by public transportation with a ruined fortification on its top and a small castle on its slopes. Alternatively, you can go there by boat; there are several companies providing trips up and down the Rhine. The hill is serviced by an electric train, so its top is accessible even if you’re not in great physical shape. From the top of the hill you have a great view of the surrounding area.
You can stay in either Bonn or Cologne, as it takes only 30 minutes to go from Bonn to Cologne by public transportation.
Bonn/Cologne -> Vienna: there are direct flights from Cologne/Bonn airport to Vienna (at least 3/day by Lufthansa IIRC, but you can check Germanwings too - they’re a low-cost Lufthansa). The airport is reachable in ~ 30 minutes from both Bonn and Cologne by public transportation (not to mention by taxi).
If you plan the trip well in advance, you can probably get the tickets really cheap. There is also a direct ICE train from Bonn to Vienna (through Frankfurt), but it takes 9 hours and probably costs the same as a flying.
I don’t know about the second part of your trip, but for the first part I don’t think you need a car, especially if you go directly from Bonn to Vienna. The distances involved are quite big, it will cost quite a lot (rental + fuel) and it may be tiresome to drive that long. The only drawback I can see is that you’d have to handle the baggage, though that’s not such a big problem if you allow enough time.
