I started four years ago with a $380 bike shop hybrid, a helmet and a backpack. I added accessories (lights, rack, fenders) as I went along. Your plan is quite doable based on my experience.

Caution is advised when cycling through interchanges with surface streets, but there's no reason to go seven miles out of your way--unless you like the ride. (I regularly extend my commute from 4.5 miles to 16.5, just because.) I have to cross an expressway and a river between home and work. Both are snarly traffic chokepoints. Both can be traversed safely, and I do so twice a day. Granted, they made me very nervous when I started out. I still respect them, but I no longer fear them.

Check with your library to see if they have the book, The Art of Cycling: A Guide to Bicycling in 21st-Century America by Robert Hurst (originally published as The Art of Urban Cycling: Lessons From the Streets). It's chock-full of tips and techniques to keep you safe when cycling with traffic.

PS: Don't sweat your first bike purchase, especially not since you're already planning to replace it in a few months. The purpose of your first bike is to teach you what you want and need in your second bike. Therefore, there can never be a "right" first bike. Just find something you can afford, is reasonably comfortable and isn't likely to fall apart this year. Then let it teach you.