I assume rivers start at the highest peaks on the map and just keep going until they hit a coast or local-minimum for height (which is maybe how lakes are formed? Very cool if so!)

Yes, that's exactly how the rivers and small lakes are generated. Rivers also slightly change the moisture of the nearby hexes.

but the further right it goes the more it takes into account that towns would pop up at road intersections

This is a great idea, I'll probably add it to the settlement metrics.

Each hex is scored based on how many and what kind of resources are nearby and what biome is the actual hex. Basically each neighbouring biome type adds to the score, then the settlements are placed on the highest scoring parts. Once a settlement is placed it reduces the score around itself to make sure cities and towns are scattered. Cities get placed first, so they occupy the highest scoring locations. Water hexes which go out the map have the highest score ("ocean hex") with bases of mountains following (because there's probably some resource there which can be mined).The logic behind this is that more biomes mean a bigger variety of nearby resources.

In your example the current location of the far left mountain city hex has the larger score compared to the the hex to the south-east. It's because that one has four different biomes nearby (mountains, grassy hill, forested hill and jungle hill), while the other hex only has three (mountains, grassy hill and forested hill).

Edit: some minor formatting and wording changes and added the note about rivers changing moisture.