Your explanation is perfect so don't worry about that.
There are many ways to tackle this problem..
I would start by setting the Pads to send the messages that are used by Drum Machine Designer.
Use the MPK mini Mk2 Editor to assign the appropriate notes to your pads.
The MPK mini uses two banks so set bank A's first row to:F#1, G#1, A#1, C#2.
Second row to:C1, D1, C#1, D#1.
Bank B's first row to:F#2, G2, A2, B2.
& second row to:D#2, F1, G1, A1.
Send it to Program 1.
(Drum Machine designer has a second set of its drum grid. You can use the same method described here to setup the second one as Program 2)
Now you are set for the Drum Machine Designer.
Regarding other instruments this will work mostly fine within Logic but there are many times you would want to change the mapping.
Drum Machine Designer makes it easy for you to reassign different sounds to each pad from its plug-in window but EXS24 or Drum Kit Designer aren't as versatile.
You can then easily do this by using a free MIDI FX script. Logic's own Drum Kit Designer Remapper preset is probably ok.
You can also download RemapNiño from unheardofski here (scroll all the way down): Free Logic Pro X Scripts - LogicScripts - MIDI fx plugin scripts for Logic Pro X
Hope that helps.