Does the local weather affect the pokemon types that appear?

The location is very important:

... what Pokémon you can encounter depends largely on your location. Walk near a beach, and you’ll find more Water types, for example.

(from here)

I cannot find anything that relates weather to the Pokemon types. I don't think it is an influence factor.

I am personally confident that weather plays a part in spawning Pokemon, through personal experience.

To quote the official FAQ;

Some wild Pokémon appear only in certain environments and climates. For example, some Pokémon may appear only near lakes, oceans, or other bodies of water.

At first, I did not read much into it. However, we have had some considerable rainfall during the last two days, and there was a noticeable difference. Instead of my street being filled with Doduo, I was finding Polywhirl and Staryu. In contrast, not a single water Pokemon has appeared in the previous few days leading from the release. This includes in a small area, nearby, that is misrepresented as a small lake.

Ultimately, it is not difficult to gather weather statistics, from a development point of view. Especially on a phone, which already has such information 'localised' for weather applications.

Do the area determine the probability of the pokemon types?

This is what the Pokémon Go description says on the Google Play store:

Certain Pokémon appear near their native environment—look for Water-type Pokémon by lakes and oceans.

I live in a big city, with no real rivers, lakes or big water sources.

I have found many water pokémon and I haven't been around a water source at all. Not even fountains. So far I have caught Krabby, Kingler, Goldeen, Staryu, Horsea and Squirtle (besides the starter one).

So maybe some pokémon will require you to move to a lake, or a river, or a volcano, but not all of them.

I'm still not sure on how much effect the location has on the game and the pokémon you find, but the game just launched 2 days ago in NZ and Japan. Maybe the pokémon we (players outside those countries) find will be more location accurate once the game is officially released globally.