Sorry that I reply late as I was pretty busy advertising both my melodica and my ukulele facebook page recently.
1. If I play two hand, no I don’t think it is possible to do it without the tube. But it is true that it sounds a bit better by just using the mouthpiece if you play one hand, I first started by playing one hand as well.
2. For my Suzuki Hammond 44, there are two pins on both ends of the melodica, a basic guitar strap would do the work. For my Mylodica and yamaha pianica, I got a strap from some sport store, and tied it on the hand strap behind the instrument. Pianonymous even has a blog in Japanese describes how she made some changes on Suzuki M37C to install the strap, but I don’t speak Japanese so I cannot understand what she said on that blog haha.
3. Well I tried playing without a strap, sitting down like this:
The strap to me mainly is for me to be able to play in two hand by standing up, hence it is easier to control the breath. To me, the better management of the diaphragm, the more natural vibrato (a.k.a. nicer tune) the melodica can produce. Vibrato really takes the melodica playing to a complete different level, it is definitely worth practicing 🙂
“the chords played with the left hand were at a lower volume than the melody with the right hand.” – Yeah it is hard as all the notes are produced by the same stream of breath. For me the way I try to do is that mostly I do chords with only two notes instead of 3, and I don’t hold the chord too long, for the right hand I mostly do the melody with 1 note, that way I think the melody comes out the most while the chord is not too overpowering the melody. You have to deal with similar issue too if you play accordion 🙂