Can 1 tuner do multiple channels on 1 transponder?
Yes - while I've never personally deliberately tested it, I've read elsewhere that channels on the same mux should be decoded by the software in parallel. I'm sure I've recorded multiple BBC HD streams on a single tuner, though.
Can the stream output in UDP so it can be picked up in another platform (non Android)?
There's nothing Android-specific about tvheadend - it can pass the stream to tvhguide (the Android client you're probably thinking of), XBMC (on any platform you can imagine), and VLC (ditto). All you need to do is lock onto the right stream... I believe that this is HTSP, so TCP instead of UDP, however. I think you can use VLC as a proxy to re-broadcast as UDP if you really need to.
What are the recommended requirements for running 4x4 tuners outputting 16 channels?
Depends on your system. USB tuners tend to meander between different device numbers on a reboot unless you take steps to lock them down, so internal tuners are probably preferred - although I use PC-TV external tuners and they work well.
TBS does some good tuners, but the drivers don't play nicely with many non-TBS devices - there are many discussions. Indeed, there's a whole thread here dedicated to "what hardware...", so that'd be a good place to dig.