Okay, I'm going to be brutally honest.
First caveat: anyone who isn't tone deaf can learn to sing (and sing well). Second caveat: musical theater is something I know lots about -- I've worked in it professionally, I write for it, and I've been musical director on a number shows. In other words, I really do know from where I speak.
To answer your question, no, right now, you can't sing. You have absolutely no technique whatsoever. Your voice isn't supported, you're breathing wrong, and your phrasing is just awful. The good news is that you are not tone deaf, notwithstanding some real difficulty matching pitch.
The solution for you is proper vocal training with a legit voice coach. All of these skills can be learned, and learned fairly easily (though it takes practice). If you want to sing, you'll be able to. Learning proper technique is just a skill -- it's craft, and anyone can acquire it (if they're not tone deaf, and you're not). What can't be learned is a passion for the music and, from your post, I'd say you have that (I love Idina Menzel as well -- she's an amazing performer). Find yourself a good vocal coach -- one who teaches legit voices, not pop singing -- spend a year with him or her and then come back to us if you want feedback. At that point, however, I don't think you'll need the feedback, and you'll be well pleased with your voice.
