(c(mod p)) (mod 2) = (p * q + 2 * r + m (mod p)) (mod 2) = 2r + m (mod 2) = m

This breaks if 2*r > p. Even if you choose r to be small during encryption, the r values accumulate with each homomorphic operation and will eventually be too big. The only restriction stated is that r is "from a different interval than the private key one". This should be made more clear.