There is normally a rubber washer between the adaptor and the compressor fitting. However, I just checked my Badger hose, and it doesn't have one! The end of the adaptor is domed. I guess the idea is that the domed brass adaptor will be compressed slightly onto the compressor fitting, kind of like an "olive" on a plumbing fitting. This type of fitting needs to be cranked up with a spanner (wrench) to make a good seal. Hand tight will leak.
I don't actually use the Badger braided hose as I find it too heavy. I use a vinyl hose instead, which does have a rubber washer.
The thread sealing tape is actually PTFE tape, a very thin white plastic tape. Used for plumbing fittings. Use it on every metal-to-metal fitting, even tapered ones. The only time you don't need it is if the fitting has a rubber washer.
Cheers