I would have no problem applying for a junior dev position, but the position they're advertising isn't for a junior dev. But you're right on just applying (read: being more confident). I am not sure I'm quite ready to forsake pursuing my degree, however. That's probably dumb of me.

About the proposal, the resources I found with a quick search were mostly for companies seeking interns. Do you have a pointer to some resources for proposal-writing for interns? (This is a new concept to me.)