I apologize for not breaking up my reply in smaller paragraphs. I'll try not to let it happen again. Now more to the OP:
Concentrate on practicing Clarke's Technical Studies (which Jon Faddis recommends for building range); Colin's Advanced Lip Flexibilities etc. I personally use Claude Gordon's Systematic Approach because it breaks things down into 52 weekly lessons. If you practice technique correctly, then your range can build naturally over time. Clarke's Technical Study number one will build your range if you play it softly, 16 times in one breath and expand all the way up to the last exercise. By playing this exercise as Clarke states, you are forced to use air compression etc. It is also basically LONG TONES as you are playing each exercise 16 times in one breath. Furthermore, you could play exercise number 12, then 11, then 13, then 10, then 14, then 9, then 15 etc. that is a technique many of Bill Adams students use (so I have been told by several) Thus, you are playing something somewhat higher ... then lower and so on.
I use a metronome and rest as many beats as I have just played with EACH exercise. Remember, you are BUILDING your chops, not destroying them !!
The biggest obstacle is learning how to support the air with enough force to elicit really fast air compression. Probably most have the strength in their corners (embouchure) to play high; however, they don't realize the force or support needed to attain the fast air compression necessary.
Frankly, we are ALL individuals. Everyone doesn't wear a size 12 shoe. It's no different with mouthpieces. Find a mouthpiece that affords you with great tone FIRST and stay with it. I wouldn't let anyone (regardless of their status) tell me what equipment to use as YOU must determine what is comfortable for you.
Shallow and Deep (mouthpiece sizes) are absolutely RELATIVE !! If you have a very small mouth and very thin lips (as I do) ... a Bach 3C is a huge mouthpiece! My Monette Prana BL2 S3 is very comfortable to my chops; however, most would classify it as a shallow mouthpiece. As I have almost zero lip intrusion into the mouthpiece, I can utilize the entire cup and not fill part of it up with lip. In my opinion, that is how so many artist are successful in playing small pieces ... zero lip intrusion and thus no "bottoming out" (where your lip swells into the cup and cuts everything off.
Building range takes some time. Be patient, relax and enjoy the journey.
This post lovingly broken up into small bites.