The first half, before everything gets intense, consists of mostly slow, balladish songs. Most of the faster music comes in Act II, when all the intense stuff happens.
It's not foolproof, but maybe defecation isn't that big a deal because it is both easier to hold in and doesn't happen as often per person.
Little Sally at the end of Act 1 points out that, considering a little girl like her has been given a lot of lines for a musical, there is still hope for the musical to have a happy ending, in spite of Lockstock insisting it's an unhappy musical. But Little Sally is generally played by an adult, or at least someone the same age as the rest of the cast, implying that all bets are off.
Urinetown is actually just getting thrown off Cladwell Column, the highest building in town. Not only is this a very public "secret" method of execution, by the time Officer Barrel shows up with his bucket and shovel, Bobby has time to sing a whole song.
It's possible that the Urinetown execution method is usually more discreet, but the UGC chose to throw Bobby off the building in order to make an example out of him.
Except that "Cop Song" basically says that the standard method is throwing people off the building. Keep in mind the UGC building is far, far away from the poor section of town. The rich people, who have plenty of money, are probably used to seeing poor people thrown off the building all the time.
It's all but outright stated that Bobby is an outlier in how long he manages to survive after the fall; Little Sally even says that his survival (as long as it lasted) was "a miracle". Presumably, everyone else who was sent to Urinetown ended up being killed by the impact of hitting the ground, and it's likely that Little Sally was only there to hear Bobby's last words because she was tailing him to the meeting with Cladwell to report back to the others if anything bad happened. It's also worth pointing out that Old Man Strong was sentenced to Urinetown in the early morning and yet it's implied in the brief scene leading up to the Cop Song (which takes place late that night) that he's only just been thrown off the UGC building; maybe the executions usually happen when it's too dark for most people to see but Cladwell wasn't willing to wait when it came to Bobby. As for why Bobby survives even for a short time when presumably no one else does? Well, he's a main character of a musical and main characters in musicals tend to get one last song before they die...
