How do urban areas affect rain?
In a study conducted in Taiwan Impact of the Urban Heat Island Effect on Precipitation over a Complex Geographic Environment in Northern Taiwan (Lin et al. 2011), found that by observations that


  The UHI effect plays an important role in perturbing thermal and dynamic processes; it affects the location of thunderstorms and precipitation over the complex geographic environment in northern Taiwan.


Hance, the situation and potential models of the distribution of rainfall are further complicated by the surrounding topography.

However, NASA's Earth Observatory page The Impact of City Landscapes on Rain presents 3 hypotheses for urban effects on rainfall, paraphrased below:


As you mentioned, the Urban Heat Island, forms a bubble of warmer air and potentially creates atmospheric instability, resulting in



  as the city-warmed air rises, it cools and forms rain-producing clouds that soak the area downwind.



The buildings and other structures cause a disruption to the ambient air flow, resulting in



  the city’s buildings provide a source of lift to push warm, moist, surface air into the cooler air above it, where it can develop into rain clouds.



Similar to no.2, the buildings split the flow of air around the warmer city, these parcels of air return together on the other side:



  When the two halves of the storm come back together downwind of the city, the air is pushed up like the two colliding trains. The rising air forms rain clouds.


When I did studies of the effects of aerosols (particularly particulate matter) in urban environments, there were many observations to suggest that some particulate matter could potentially become the cloud condensation nuclei that changes cloud cover and hence rainfall patterns.