Within Europe, the general answer is yes, you can drive your hire car into a different European country from where you picked it up.
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Many car rental suppliers will allow you to travel outside Germany without charging additional car rental fees, although there are specific cases in which restrictions or charges may apply. Most commonly, traveling to Eastern Europe with a rental car out of Germany has very specific requirements for this to be allowed.
Most of the time you'll pay a cross-border fee to take the car to another country. However, some car hire companies, such as Europcar, offer deals that include cover for the car abroad, so you won't pay extra when you arrive.
Renting in Germany is usually cheaper - but if you drop it in a different country - there is usually a drop-off charge - which may be significant. (Shop around before you organize your rental). Also - Alsace is in France and Geneva is in Switzerland. As nytraveler says, Germany is usually the cheapest option.
Out of all the popular cities in Europe, Bucharest is the cheapest city to rent a car in at just $2/day. Renting a car in Bucharest is 85% cheaper than renting one in Dublin where the per day rate is around $13.
Rental cars are slapped with Germany's 16% Value Added Tax (VAT), registration fee, and airport fees (but rates do include the required third-party liability insurance). These additional fees may equal up to 25% of the daily rental.
You must have third party insurance cover (a legal requirement) at the least when driving in France, but it doesn't cover any costs incurred by you as a result of an accident. Make sure your policy is fully comprehensive and that you have your Certificate of Motor Insurance before setting off.