You can bring as much "hand luggage" with you as you can store under and over your seat, and one peace of "carriable" (by one person) luggage which can be stored somewhere in the aisle (your responsibility).
For strollers the best places are generally at the beginning and end of the car where the seats can be fold up. It's advisable to reserve such seats, ask at the ticket counter.
DB outsourced luggage transportation in 1995. Which means that "checking in luggage" is really "sending them as parcels with the Hermes courier service". You might want to compare prices with other courier companies, DHL and UPS tend to be cheaper than Hermes, especially with heavier/bigger parcels.