1. Are ADF's sensitive to water currents like ACF's are or is a regular aquarium filter ok to use?
yes they are but you can just get a whisper filter they dont provide much current ... There is some controversy on sponge filters

2. How do I decide if I should add frogs to my tank that already houses mollies and loaches or set up a new tank devoted only to ADF's?
Usually a species tank is recommended however you can usually get away with adding them into a tank if they have places to hide and you are making sure they eat.

3. If you have a filter in the tank, how do you keep a lid on the tank with no spaces for escape? The aquarium filters I have always used hang over the side of the tank, thus creating a space that isn't sealed in by the aquarium lid.
I personally use an in tank filter and just put something over the intake to keep any fins from being sucked in

4. Are ADF's able to transmit disease to my fish or to catch disease from them? For example, if I had frogs in with my fish and then the fish got ich, it wouldn't be possible to treat the fish without first removing the frogs, correct?
No frogs are amphibians and they can not catch or transmit anything to your fish.

5. Does the tank you are going to be putting the frogs in need to be completely cycled (the nitrogen cycle) or since they also come up for air, is the cycle not important as long as regular water changes are made?
You must cycle your tank ..African Dwarf Frogs do not drink they absorb moisture through their skin so if any ammonia or anything in the tank they are highly sensitive to it.