I'm looking for a combination of drum machine for live music that does not require a PC, but will also serve as a MIDI controller for various DAWs. Is there a name for such a combination? I'm not looking for recommendations, but simply what that combo might be called, along with an example or two, good or bad.
I suspect you haven't looked very hard yet because there are lots.
There are cheap but not bad ones like these: http://usa.yamaha.com/products/musical-instruments/entertainment/digitalpercussions/dd_series/?mode=series#tab=product_lineup
and more expensive digital drum kits http://usa.yamaha.com/products/musical-instruments/drums/el-drumkit/
I like the Yamaha products, but there are plenty of other manufacturers. (I have no affiliation with Yamaha)
Almost all have MIDI outs and a built in synth that'll often have melodic patches. But drums can be messy across MIDI with the "which dang note is which dang sound" mapping thing and program changes and such.