Tips? Nope. Well, push from your diaphragm and if you're into any wind instruments at all, practice long tones. On the melodica it doesn't matter as much, but you should still be opening up your throat as much as possible to get max airflow, like if you were to sing "properly" and not use your belt voice. Other than that, I do tend to play it (finger it) like a sax because my piano skillz are seriously lacking.

I know what you mean about letting the breath do the work to impart flavor as it were. For me, this is weird because on a sax it's all about bending blue with the mouth and moving as much air as possible, which means expanding the neck. The only real tip I can offer as a (former) wind instrument player is that you really have to use that spit valve and get all the spit out of there or it will get gross. You might want to research the plastic it's made of to see if it can handle rubbing alcohol to flush it out once in a while.

Anyhow, I love these things. They're so fun and have so much character. I'm sure that someone who's a multi-instrumentalist (like you!) can figure out this thing way better than me. I mostly bust it out to play some standards once in a blue moon.

If you go to enough jazz concerts, you will see EWIs. I haven't played one, but I'd love to someday.

As for the horn/keyboard thing, I find that I almost never play chords on it, which is under-utilizing it for sure, but I'm not yet used to being able to do chords while blowing or hell, even simultaneous melodic lines.