score:4
You have a lot of questions, which essentially depend on your own preferences and details in your planning, which you don't share with us.
What kind of bicycle you prefer to ride, a city bike, a touring bike or a mountain bike is something you have to decide upon. What makes you think that a mountain bike will be more suitable for a ride around MΓΈn than any other bike? If you are staying on roads (be it regular roads or gravel roads) and you are not planning to go off road, there is usually no need for a mountain bike, but that is valid everywhere and not a particular advice for MΓΈn. Mountain bike wheels are built to give you grip on rough underground and are (much) harder to pedal on roads than bikes with wheels made for using on-road. Remember also, that mountain bikes usually don't have luggage racks.
There are many bicycle rentals in Copenhagen and easy to find rentals everywhere else in Denmark. Almost without exception, it is quite a bit cheaper to rent for a longer period than for many shorter periods. Just to compare prices at an arbitrary rental in Copenhagen, they charge DKK 125 for one day, DKK 225 for three days and DKK 495 for nine days for the cheapest category of bikes. But, you must also remember that it is usually not free to take bikes on the train in Denmark. For example from Copenhagen to Vordinborg, you pay DKK 36 for a bike ticket if I understand the price correctly. If you plan to travel a lot around, it may still be cheaper to rent at different locations for shorter periods. Also remember, that you must usually return the bikes where you picked them up. I've seen chains of rental stores in other countries offering to pickup and return at different locations, but I don't think that is common in Denmark.
Vaude panniers are indeed not compatible with all luggage racks, but if you go to a larger rental and tell them when booking that you need such a rack, there should be no problem for the bike rental to arrange that for you. Many bike rentals also offer panniers for rent if that is easier for you than to bring your own.