Wouldn't confessing after Mass make receiving the Eucharist before making that confession a sin?
Not in the Catholic Church. We believe that one can receive the Eucharist as long as you are not in a state of "mortal sin", but one can go to confession even for just "venial sins". So if one is not in mortal sin, he can receive the Eucharist at Mass, then go to confession afterwards to confess his venial sins.
That being said, it does seem backwards to me as well - I would much prefer to receive confession BEFORE receiving the Eucharist, not afterwards. Unfortunately, I think the main reason that confessions are heard after Mass is due to convenience; the time can be more open-ended, to allow for more people to go. For example, if Mass is on Saturday at 9:00am, and you start confessions at 8:00am, then there is a hard cut-off of 1 hour. However, if you have confessions after Mass, then they can go on for as long as people are in line. And before anyone makes the comment, the long lines do exist in many Catholic Churches (but not all, unfortunately).
Also, the Catholic Church does recommend going once a month for confession. Just this weekend, my daughter received her first confession, and the priest mentioned the monthly recommendation in the prayer service that was held. However, only once a year is required. Since I don't know the habits of every Catholic in the world, I can't comment on how many Catholics receive monthly, and how many receive less often.
Personally, my wife and I have set up a monthly appointment with our associate pastor to hear our confession. This has been wonderful to my soul. (Also, I have to admit: it would drive me crazy waiting in line for over an hour to have my confession heard during the "regular" time - I would inevitably have to confess this!)