Yeah, these opportunities are all contextual - each city district gets given a single randomised bonus feature, and each of those can spawn certain of these events. The current bonus features are temples, heroic statues, ruins, tunnels, barracks, fortifications, alleys, marketplaces, tanneries, workshops/smithies, libraries and schools (as you can maybe tell they kind of come in thematic pairs). I may add more bonus feature options at some point but I think this list is plenty to be getting on with! Some districts may spawn events based on other features too - for example, there might be some events tied to a district being really poor or really wealthy.
There will be limitations but I'm not sure exactly what, yet. I think mainly I'm going to make searching take a day of game time, much like staying at a tavern does (the game doesn't actually use days as a metric, it's awkwardly calculated by dividing the length of a week by seven and then rounding so sometimes you get six or eight day weeks, but meh). So that gives you some limitation. There should be more limits on things like cleric hiring so you don't end up getting spammed with unemployed clerics, but I haven't worked those out yet.