I can't see you getting a car for much less than €45/day, €60/weekend or €200 per week. You *can* drive in Frankfurt: traffic is generally not that bad and parking is possible - but budget €20/night in addition to the cost of the car. The city is fairly walkable and public transportation is cheap.
I've stayed at the Moxy in Frankfurt centre - and hated it, by the way. Incredibly disorganised staff with very basic rooms (not even an iron and ironing board - fatal to a business traveller!)