If you assume that your monthly returns are independent from each other, then the annualized variance of each series, and the covariance can be annualized. This assumption allows you to use V(x1+X2+...+x12) = V(x1) + V(x2) + ... + V(x12) where xi is the return for the month "i". Actually, for this to happen you only need a weaker assumption: that is that interperiod returns correlation or covariance be zero since V(x1 + .. + x12) = Sum(i=1..12,j=1..12,Cov(xi,xj)).
Then if you add the "identically distributed" assumption which means that x1, ... , x12 are just the repetition of the same experiment and follows the same probabilistic law: you get in particular
E(x1) = ... = E(x12) (same expected returns)
V(x1) = V(x2) = ... = V(x12) (same variance)
Finally, V(x1 + .. X12) = V(x1) + ... + V(x12) = 12 * V(x1)
That is: V(annual returns) = 12 times the variance of monthly returns.
Beta and R² are already "normalized" so no need to "annualize" them. Under the same assumptions, you are trying to explain one series of returns with the other using a linear model. Whatever the relation of monthly returns, you will have the same on annual returns.
Final remark: not that assuming independent expected returns means that your monthly returns have no memory. But sometimes, they do: the returns from month i and returns from month i+1 are correlated (see Markov chains for example).