Media types have heavily scrutinized and occasionally criticized the NFL for allowing its players to behave irresponsibly (e.g., to get arrested for DUI, or to “make it rain” in a strip club). Even during the season, it seems rare that a week goes by in which one or more NFL players wasn’t arrested (see the “PFT police blotter” on profootballtalk.com for a list of arrests). How a team acts is at least partially a reflection of its coaches and partially a reflection of the strategies used to put the team together (drafting and trading strategies – usually of the general manager).
A popular hypothesis floating around these days is that the Cincinatti Bengals were willing to overlook character flaws on draft day if the players they drafted had enough football ability. It’s easy to show that the Bengals had a lot of high-profile arrests, but it’s much harder to say that those arrests indicate that the team is intrinsically different from any other NFL team. The Bengals might have just had bad luck, or the Cincinatti police department may have a particular grudge against the team.
A well-coached team will generally not have many arrests or character issues (which is obvious). Nevertheless, any cross-section of 60 or so (including practice squad) rich males in their 20s and early 30s seems likely to encounter at least one DUI, if not more, in the time that it takes for a football season to elapse. So as far as a metric of how well coached a team is, I don’t think arrests by itself is very useful. It includes too many random elements.
A possibly useful metric, though, is how a team performs at home vs. how it performs away (controlling for confounders such as domes, unique wind patterns, particularly loud stadiums, etc.). When a team goes on the road, its players are isolated from their friends, families, and favorite bars/clubs. The team goes to a hotel, and typically has an enforceable curfew. At home, on the other hand, there is a lot more freedom for players to get into trouble, or to at least have a few drinks/stay up late, thereby affecting performance on Sunday.
It might be informative to look at home vs. away performances differentials of different coaching regimes for the same team. Compare, for example, the Jim Fassel home vs. away differential to the Tom Coughlin home vs. away differential. That would control for confounding, stadium specific factors (because they’d be relatively constant across regimes).