You are an excellent musician, I can tell. I would be willing to guess that you play piano or some other instrument quite well, and that you have a very good choir director. You have a very good sense of pitch, rhythm, and possess VERY clean diction! =) I can tell you are very intelligent... because the vocal problems that I hear are usually brought on by over-thinking! ;-) You have smart person problems.
Your tone needs a lot of work. I recommend that you get a voice teacher to guide you; no one can do this alone.
You are very breathy and are not supporting with your gut at all... sounds like it is completely in your throat. I find that this usually happens when a singer thinks she is breathy and tries to fix it... the wrong way, with the throat. Loosen your throat, but tighten your lower abs when you sing. HARD. You have to learn to trust your throat, quit listening to test if you sound good, and just let your vocal chords to their thing without any help from you.
Your (rare) pitch issues are vocally related, not ear. You push with your throat and that sends you sharp. So, release your throat & trust your ear.
Exercises I assign for this are designed for the long term, often two years or more. That said, here is the long term plan:
Learn to access your low chest voice but never sing full chest above a middle D or so (having a chest voice will add resonance... but please don't take this as permission to start belting, which would just replace one problem with another, worse, one).
You won't be able to have a free head voice until you learn how to use your gut. This is because your throat is so tight that it pinches off both your air and the ability of your vocal chords to do their work, which they can do perfectly fine without any help from the neck.
You may be supporting, but if you are it is MUCH MUCH MUCH too high in your abdomen. Start your support at the bottom of your gut (yes, yes, I know where the diaphragm is. Sing from the bottom of your gut anyway.)
Sing lots and lots of scales on a lip trill up and down the full extent of your range (AFTER A GOOD WARMUP).
If you can't turn your head freely from side to side while you are singing, your throat is in the way. Learn to let it go.
Learn to manage your breath, it is leaking out everywhere but where you need it.
Good luck to you!