I am a 16 year old with hardly any experience in the academic field and have shown a certain degree of unprofessionalism in the initial draft of my first paper. Should I state this in the cover letter and probably apologize for that?
No, you should not mention your inexperience. It's simply not going to factor into the editor's decision as to whether to publish your paper. Papers are (at least in theory) published purely on their own merits. It doesn't matter whether your paper is a good paper for a 16-year-old, it matters whether it is a good paper, full stop. So it won't get any special consideration.
When you use the term "apologize", it makes it sound as if you are thinking: "I'm sorry for wasting the editor's time". If that's the case, just don't submit it. If you submit the paper, you think it's worthy of their consideration. It doesn't have to be perfect.
Note that a number of journals have special awards reserved for excellent papers from young/early-career scientists. If this journal is one of them, you may want to advise them of your age if/when your paper is accepted, but not before, so that they are aware of your eligibility.