Just wondering if the Boss DR-880 can output multiple streams of midi data. Say, a stream for the snare, a stream for the kick, a stream for the toms, a stream for the symbols. 

Would like to be able to record a midi pattern from my 880 and apply separate processing over each piece of the midi 

If you are using SONAR you can break out each instrument after recording the MIDI track by running the "Split Notes To Tracks" CAL script. Many don't bother with breaking out the MIDI choosing to have the synth that is producing the drum audio send each part of the kit to separate audio tracks. Of course, you can do both if you wish.

I copied all my old trusty 505 patterns into Sonar as I find some of them handy for scratch tracks. 
I used MIDI sync.
I find the Fastest way to separate the parts is in piano view to click the corresponding  key on the left which will highlight all those notes. You can copy/ paste/ delete etc. 

Cal scripts will work too.