The bad: 10+ yr old DUI & Driving under suspension, a couple underage consumptions, and medical/license revocation 5 yrs ago for not reporting the DUI. Big mistake, I know. All these incidents occurred before I began flying, and are over ten years old now. This past ten years: a few minor traffic violations (speeding).
Your past is more than a little checkered. It will hurt you.
You need to move beyond your past, and be able to show that you've outgrown it. Of all the items you cited, the medical and certificate revocation is the most glaring.
You indicated that all the incidents occurred prior to flying. Obviously not the certificate revocation. It appears that you're saying that your pilot certification was revoked, along with your medical, for not reporting driving history events that occurred before you were a certified pilot. This is something you can explain, but you've got to get the chance.
You're probably going to need to get with a regional and move up, if one will hire you. If you can get a solid history behind you, later you'll have the option to apply. As you get a year down the road, those events fall a year behind you. It will take time to clean up your background.
You indicated that you've continued to have driving infractions over the past few years. You seemed to minimize this by saying they're speeding infractions, but you've got to understand that employers are loathe to put you in their twenty five or fifty million dollar airplane when you aren't showing good judgment with your own car. Particularly in airline flying, the ability to adhere to profiles, limitations, regulations and policies carries a lot of weight. Your history shows a lot of recklessness, and that includes the speeding infractions.
Many people get an occasional citation. It's not the end of the world. When you add up multiple citations, however, as well as driving under the influence, and FAA certification revocation, you've got a formidable past which may haunt you for a while. Show that you've learned from that, and become Mr. Squeaky Clean. Not a speeding ticket. Don't even fart in public. Your career is far from over, but you're going to need to build yourself back up to a more marketable (and respectable) position.