Boots,
Are you writing the drum sounds back to tape? i.e. Are you planning on sending the outputs from the drum machine to individual tracks, or are you just wanting to use multiple outputs during the mix?
I ask because if you're planning on writing the drum tracks to tape, you could accomplish the goal using the existing drum machine and just making two passes, the first pass being snare and kick panned hard left & right, and the second pass being the rest of the kit in stereo. If you're not writing to tape, then yes, you'd need something with more outputs.
The Alesis should sound fine. It's not in the same league as a TD-20, but the main difference the TD-20 would bring would be noticed if you were triggering them with V-Drum pads. If you're simply programming the drums with a sequencer, you probably won't notice much difference. The Alesis machines are solid, and are best-in-class when it comes to a budget machine with flexible output options and decent sample sounds.
For your MIDI I/O questions, I think the DR-5 will output it's MIDI sequence just fine... I checked the Boss website, but the only support documents I could find were the turbostart manuals - no MIDI charts. If it works like most drum sequencers, the MIDI for the drums should all be on channel 10, so you'd just set up the Alesis (or whatever you choose) to R/X (receive) MIDI on channel 10, and it will play back whatever data is transmitted by the DR-5.