Hey potus2080, you can put a compressor before or after the MP1 but depends what you want it to do (before needs to be instrument level, after needs to be line level). Short answer I'm with MJMP (why except the gate (which is definitely better after MP1 and good to use only if you play really high gain to tame the MP1 as high gain = high noise (the noise is boosted too))). But this is all very subjective, you should do what sounds good to you...
My basic premise is to keep the signal simple and analogue all the way from Mp1/2 to poweramp/cabs and "mix" into that signal what else you want (e.g. digital effects).
Longer....and my "(quite pedantic) opinions"
MP1 input - I wouldn't put any thing in between the guitar an the MP1 (except a good guitar cable, or radio thingy if you must...(I'm not a big fan of these BTW...)). Let the MP1 do what it's designed for, take a good guitar signal (millivolts) and turn it into volts with a gorgeous tone.... (also adds eq, stereo chorus and compression (if needed)), I wouldn't put a tuner in between either unless it has a "true" pass though (or it will change the tone (albeit subtlety))
The Alesis (from what you've said and what I quickly googled) is a compressor and a gate (and stereo, designed for PA use I suspect, not guitar ??). In your set up, IMHO you don't need it as a compressor, the compressor in the MP1 is fine (I get your previous patch using each channel but was with a mono "head", your MP1 is quite different, so I'd leave that idea behind, good idea for those bits of kit, not with the MP1. The Mp1 is not a "head" it's a pre-amp (1/2 a head ? and quite different in many ways to the pre-amp in a head)).
You can get compression in 2 ways in a MP1 (3 ways if you use the loop), first by cranking up the gain and making it distort (basically what it's designed to do) and second by turning on its compressor (which I only use for clean sounds BTW). Overloading tubes (clipping/tube distortion) also compresses "overloads" the next stage. You don't need a compressor before the MP1, it will just hamper your picking dynamics and suck tone (the tubes do it all better).
FX is harder with MP1 because you want to keep stereo (so not use the mono loop (except for mono things) and it's serial, don't put digital gear here if you want to keep good analogue signal) or 4 cable method (i.e. chain units together, MP1 A/B out to Eff L/R ins, Eff L/R out to power amp > cabs) and the first thing most digital stereo effects units do is A/D your signal (so now the signal isn't analogue anymore (IMHO, not good!)) better to mix the digital effect(s) with the MP1 analogue signal (use a mixer (which is what I do) or the Art Split mix (or similar) works also but not as versatile. Again, each to their own, but it does sound better this way.
I'd trade the hush and BBE on something more useful, like a good quality small analogue desk with ~8 channels and 4 effects sends (not so easy to find). Check (http://adadepot.com/index.php?topic=117.msg391#msg391) for more.
The hush is another noise gate for super high gain playing (from what I can tell), helps with bad gain structure is another view.
The BBE (from what I can glean) is like an aural exciter for guitar, adds upper harmonics (6khz, 18khz et al) makes you stand out in the mix (I'd loose it and turn your presences knob up = +6 khz).
Anyway my rant, hope it helps Cheers R