
There's nothing wrong with lamp's answer but I think you're looking for more clarity on what a MPC can do.
A - YES: of course, assuming the synth has a MIDI IN jack. It won't control something that's only CV based (if you don't know what that means don't even worry about it).
B - YES: Standalone MPCs are samplers that have some sort of audio input. Whatever audio you feed it, it can sample. That audio could be you playing your synth, in a pattern. That's a big chunk of audio locked to the tempo you used when recording. If you change the tempo the pitch will also change.
If you hooked your synth up through MIDI, then you record a track on the MPC and the MIDI notes get sent to that synth. The audio in that case is actually still coming from the synth, which is fine if that's what you want. That's Lamp's reply, but technically, you can do both.
You probably want to learn how to do both these things so you can do whatever you want when you want it instead of being stuck with only one workflow.
C - YES: if the MPC supports it you can usually assign a sample to a pad and then do some sort of 16 Step mode on that sample. If that sample is a single note being played off the synth, then you can spread it over the 16 pads, each step being a chromatic note. Then you can play the notes back as a 'pattern' and record that pattern into a sequence.
YouTube is your friend, go look at some basic workflow tutorials for the MPC you are interested in to get a better idea of what they are capable of.
Hard: MPC2500, microKorg, microBrute, miniBrute, RE20 Space Echo, Patchblocks, Akai Push2
Soft: Ableton Suite, Reaktor 6