In all fairness, rape usually commands significantly longer sentences than 2-3 years. Even if released, conditions are frequently imposed on the parolee--like no contact with minors, if appropriate to the circumstances. (But the feeling of uneasiness that you suggest is understandable.)
Speeding tickets
in general are not fair game for disqualifying a person. In addition to their proper use (as a penalty for bona fide dangerous driving), they are used as revenue-gathering ploys by certain unscrupulous municipalities. If you happen to get caught 4 times in three years by cops trying only to fill their quotas (implicit or explicit quotas--it doesn't matter), you aren't
necessarily a threat to society in general, or FIRST in particular. But: I'm not sure about specific U.S. laws concerning speeding, but I assume that
there exists a distinction between felony speeding and a simple traffic ticket (a felony being a full-fledged criminal offence, which is permanently recorded). In the case of speeding as a felony, it could be seen as equivalent to driving while intoxicated, fleeing a police officer or leaving the scene (of an accident) all of which are serious, and might well be grounds for disqualification. In the case of minor speeding offences (doing 2 km/h over the limit in a speed trap), there is no basis for disqualification. Actually, I think FIRST has addressed this distinction with the appeal procedure--but it isn't likely to please anybody who gets red-flagged for whatever reason, and then has to jump through bureaucratic hoops to clear themself. (The onus should not be on the mentor to justify himself for irrelevancies like non-felony speeding.)
A better question was brought up earlier: is FIRST liable for giving convicted felons (who have served their sentences, and been released) an opportunity to re-offend? And is this any different from the level of liability that would apply if any old mentor decided to do something reprehensible? And by the same token, is FIRST willing to let a rehabilitated felon give something back to the community, or will they forever dismiss and condemn them out of hand? The answer is not immediately forthcoming.
It's like the stock market disclaimer: "Past performance does not necessarily reflect future trends." You can eliminate those who have offended before (and probably reduce the risk), but I hope that nobody sees this as a catch-all solution. There exists the potential for misbehaviour whether or not the ones with a shady history have been eliminated, and no amount of background checking can eliminate all risk. FIRST is making an effort to minimize that risk (and that's a fair objective), but one wonders if they've gone a little bit too far, once again. (By "once again", I'm referring to the aforementioned crowd-surfing incident that caused dancing to be banned at FIRST team parties. While dancing is surely the devil incarnate

, you've got to let the FIRSTers have some fun.)
What it comes down to is this: FIRST is taking it upon themselves to force the teams to comply with a set of far-reaching regulations intended to protect the innocent. It's a worthy cause, but it's tarnished by a heavy-handed approach that incorporates none of the charisma that FIRST exudes in so many other pursuits. They're offending longtime and distinguished supporters, and forcing bureaucratic nightmares and liabilities on the team leaders. This is not a fair burden to place on the shoulders of the very people who so strongly support FIRST as an organization. While these measures may decrease the chances (infinitesimally) that harm may come to a FIRST team member, Bill Beatty put it best when he suggested that the world is not always a perfect place, and that people need to be somewhat responsible for looking out for themselves. I'm not blaming the victim--rather, I'm saying that we shouldn't need FIRST to look out for us; we shouldn't need to check and bianually re-check every mentor, out of the fear that they may be concealing a dark past. Take them at their word (make them put it in writing, if you must have something tangible by which to hold them accountable)--but don't treat them with suspicion, just because they haven't yet been proven innocent.
And if something were to happen to a FIRST team member, as a result of the mentor's misdeeds, it would of course be a tragedy--but let's not jump to the conclusion that a background check would have prevented that person from committing a crime. If they had been rejected from FIRST, there are plenty of other potential victims everywhere--and if one of them were victimized instead, we have no right to claim that our background checks did any good. There was still a victim.