Where do you plan to return the car? One way car rentals in Europe are still ridiculously expensive.
Do you know how to drive a manual? Automatics are also ridiculously expensive.

1. Taking a car into CZ is not normally a problem. Be sure to get a cross border card that will let you take it across the border there and to Austria to ensure that your insurance is valid in all of those countries.
2. In some German cities there are environmental zones. As mentioned above, you also need a toll sticker for Austrian and Czech motorways (can be bought on the border)
3. Unlikely to affect you as long as your paperwork is in order.

When visiting cities be sure you have a place to park.