People tend to take them at their word, but if this is abused and caught it may become something that's more rigorously checked. But Bonds or claims thereof also aren't 100% required for leaving the city - so long as you have someone with you, or are being supervised by an experienced Bonded pair or Wilder team, typically individuals are given a free pass.

Typically. Eren and Mikleo have already gotten in trouble for lying about where they were going. Further examples of this will result in a change of the rules.