Another way of doing this is via home exchange. We do this most summers for 9-11 weeks (sometimes one exchange and often two, back-to-back) and there are many people we know of who do it for many months at a time, like you are planning to do.
Just be aware of the limits of your country's passport and how long you are allowed to stay absent a long-stay visa. You have enough time to attempt to get one but, moving around as you are, it most likely won't be possible as you have to have an address in Europe.
To accomplish what you want to do you will probably need to combine Schengen and non-Schengen countries so you don't overstay.