The answer to your question is NO! The kinds of tickets you got will NOT exclude you from getting a CHL. My story is VERY similar to yours. I got arrested(more than once) for expired registration, expired tags, speeding, and theft of check under $200(bounced check from 1985 & 1987, that I took care of before I was arrested).
What you WILL need is certified copies of the disposition of each and every arrest. You can get those certified copies of the dispositions from the county clerks office where each arrest occurred.
It took me 2 or 3 days to get those dispositions, because it took me forever to find the exact county clerk's office here in Tarrant County that knew what I was talking about(they have 9 of them). I wound up at the Criminal Courts County Clerk office and explained what I wanted. In Tarrant county, they don't call it a "disposition", but a "Final Judgment and Sentence". It's the same thing though.
Once you get those certified copies of the dispositions, you'll need to send them in to DPS so they can be merged into your file. Be sure and put your PIN number on each and every piece of paper you send DPS.
I'm not sure of the procedures now, but I sent all of my certified copies of the dispositions, AND an explanation(in my own words), of the infractions I had committed and the results of each infraction. I wound up with 2 pages of type written, single spaced, in landscape form, of explanations, along with my CHL application packet. Send them the original certified copies, and you keep a copy for your records.
In addition, I got certified copies of a criminal background check from both the cities of Arlington(where the arrests occurred) AND the city of Euless(where I got several ticket), just to be on the safe side, and for my own piece of mind. (probably didn't need those, but it made me feel better)
DPS is wanting full disclosure on each and every arrest that has ever happened to you. That's what a DPS lady told me when I was fretting over those old arrests.
I hope that helps a little bit. If you have more questions, or need clarification, please do ask away!