Don't even worry about your traffic tickets for speeding. Just list and take the proof you paid them already. The officer probably won't even ask you about it but you know, there are always the detail guys out there.
Even if it is a reckless driving or DUI if you was already in the court and paid fines and any other penalty (even community service), if you cleared it won't be the root cause to deny you unless you have others citations. Problems began with things like battery, assault, sex offenses, drugs, etc. or if you really have a bad pattern for instance with DUI/reckless (2 or 3 every single year...)
Just 3 speeding in 3 years wouldn't hurt a bit your case if all is clear by now including the payment....
Hope you were not caught at 200MPH in a 15 school zone at least...
Cheers