Yes, if the keyboard or module has a MIDI IN, the Zendrum can trigger its sounds.
Just a matter of assigning the Zendrum to the proper MIDI channel.
Drums are usually on MIDI channel 10, whereas the keyboard sounds are usually on channel 1 or 3.
Yes the FAT Kat pedal is the right one to use, and you can plug it directly into the Zendrum's external input and program the note/sound it sends just like the other triggers. If thats a kick drum, chances are it will be note number 36 C1.
Hope this helps!