Don't try and use less mouthpiece, whatever you do! It will close off your sound making it choked. You should use as much mouthpiece as you can and still tongue, if that makes sense to you. Most of my students are at least halfway and more on their mpcs.

I would still have your horn checked for leaks by a tech for that low Bb, it can be tricky spotting leaks like those, because they aren't necessarily located at the Bb Key. As for the thin sound in your upper register(altissimo is triple G and higher), it's your throat, and your mouthpiece. Open your throat up, and let more air flow(not the same as playing louder). Also, a C* isn't anything for jazz. Band directors seem to be obsessed with them, but don't think they're that great for either jazz or legit. For jazz, get a large chamber piece. Try an Otto Link 6* or Meyer, they're both the same deal for the most part. Both are under a hundred bucks, and it'll have you playing with more expression, flexibility, and a fuller sound. You may not notice it at first, but as you learn to play a jazz mouthpiece, you notice a huge difference.