Kazan seems like a much more authentic and exciting option. No doubt about it! It's an historical city. I haven't been there, but I'd love to go.

I have no idea how much cheaper - but it should be significant. Accommodation in Moscow has an appalling reputation for bad value for money - it's all over the internet. Anything will be better value.

Just take the cheapest option they offer and see if you can put up with it once you are there. If it doesn't suit you, get something else once you are there.

I think you would need to provide an address for the visa but I have no idea how seriously that is checked.
Probably there is a way around that, but I wouldn't know .
Why don't you go onto a Russian social media site (Vkontakte, for example) and try to ask some students in Kazan?

Here is a link for a Yandex search for rooms to rent, in Kasan. Add student to that, as well. Ой!

Top hit is this: http://kazan.sutochno.ru/rooms/

What an exciting adventure, hope you'll have fun!