I'm not a lawyer, so take my advice with a grain of salt, but here are some thoughts...
"After this incident, I ran a criminal background check on myself from a website that charges 30 or 40 dollars that I found online. That background check came back clean. It claims to check all FBI, State, Federal, and a whole bunch of other databases. But I know not all background searches are created equal."
Any background check where you're not asked to submit a fingerprint card is not thorough enough. Most, if not all, internet companies who claim to run "extensive" background checks likely are not. These companies run name and age searches, and if your name is common enough, a "check" would return false positives.
Real background checks take time to process and require a bit of legwork if the company knows what they are doing.
SO... my questions are:
1) Was this a misdemeanor?
For Marijuana? If you paid the fine then no. What was assessed then was an "administrative fine", which was placed on the car and technically was "impounded" until you paid it.
2)Should I mention this when the program sends me the paperwork after match day?
Honesty is the best policy, really. You could not mention it, but if you don't and it comes up, the general thought is, "what else could s/he be hiding?"
2) What kind of criminal background checks do residency programs conduct? Do they query the CBP database? Does CBP even have a database?
That's the $64,000 question. Err on the side of caution and assume CBP maintains a database within the NCIC.