Alpha and beta corresponds to theta(1) and theta(2) under correlation targeting model. If you relax this assumption, then an additional coefficient (constant) will be estimated and reported as theta(3). If you choose student's t distribution, the degrees-of-freedom parameter will be shown as theta(4). And so on... You'll easily figure out with a little bit of experimenting. The footnote in the output table also refers to the stability condition, alpha+beta < 1.

Dynamic correlation models work on the variances of residuals that are obtained from the mean equations. For stationary series only a constant would be enough, but in other cases one might need to include autoregressive terms and/or explanatory variables to make residuals stationary.

Note to moderators: I kindly ask you to move this post to

DCCGARCH11 thread, where it would be helpful to other users as well.