Why is reverse osmosis desalination the dominant method used and not heat-based methods? Also why is brine being dumped back into the ocean?
Starting with question 2): you can use heat to evaporate the water but you need some sort of energy or thermal exchange to cool and condense it back to liquid. My understanding is that reverse osmosis (RO) is more energy efficient overall.

Question 1) Brine is only put into the oceans from system that are near the sea, which usually use seawater as a water source. The brine has an insignificant effect on the ocean salinity once it is mixed in but can have a local effect because of poor mixing and because differences in density between the brine and seawater cause it to sink to the bottom. Systems can be engineered to minimise the impact.

Inland, the brine can degrade groundwater resources if it is disposed to seep into the ground. Even in hot, dry climates, it may be difficult to completely evaporate it to dryness in lined basins so it doesn't seep in. Seepage basins are a viable option in some places where the groundwater is already to salty to use as a resource (for example where the salty water is extracted, put through an RO system and then reinjected to the same aquifer).

RO systems are designed so that the salty concentrate is removed during operation so the system doesn't need to be cleaned to remove the salt. However, impurities may still require cleaning of the membranes. There are ways to minimize the amount of cleaning needed.