Something like this would require the use of a DAW. There are DAWs freely available such as Reaper on both platforms well as GarageBand if you are on a Mac. There are probably more, but here are a couple for a start.
Just to clarify, the Drummer keymap is only useful if you are using BFD in conjunction with Logic's drummer. If you are not using Drummer, then use a more suitable keymap. When I'm not using Drummer, I will often resort to a modified BFD2 keymap, as opposed to a BFD3 keymap, as it is more compatible...
In order to clear out the grooves, you are best to trash the database files. All depending on which OS platform you are running will depend on where the databases are located (on the Mac, they should be in your ~/Library/Application Support folder).
Both are moving targets, so you are better off creating both BFD and maschine controller mappings to your liking. As far as triggering BFD3 with the hardware, you can create a controller map within Maschine to correspond to BFD's keymap. As far as triggering Maschine from BFD3, you can create keymap...
That does sound strange. I would suggest that the next time you encounter something like this, post a screen shot of your kit piece inspectors and possibly the mixer channel in order to see what happened here.
Equator is a synth, which is far different that BFD3. Equator's patches and presets are different from BFD's. What you need is either a DAW like Ableton or Logic, or something like MainStage to control both BFD3 and Equator at the same time.
MIDI import works for me with no issues. Did make sure that either the MIDI you are importing conforms to the key map you are using or did you make a key map that corresponds to the MIDI that you are importing? If neither match, your MIDI will not be imported properly.