Generally speaking, you have to have a conviction reported in NCIC in order to be rejected by the computer. (You can't lie on the application, though, so if you have a conviction that isn't reported in NCIC, you still have to answer the question as asked). If the disposition was something other than conviction (like supervision, conditional discharge, court diversion, nolle prosequi, etc), it shouldn't be a problem at all. Speeding tickets unless they are criminal in nature (like going 20, 25 or 30 over depending on the jurisdiction) are not problms.
Expungement is also a good option. May I ask what the two minor incidents were and what the disposition of each case was and where they took place? That might help answer your questions.
The first Was shoplifting as a kid, which I know is 'sealed' but always pops up on through vetting checks I've had to do. The second I'd rather not mention on here but I got a stay of imposition which I believe was that nothing would go on a public record if I paid a fee. Which i did. And of course like 4 speeding tickets, 1 was dropped I am fine just skipping GE as none of these have been issues in the past but if i can expedite customs at all I would love to.