Stress tensor in product of 2D CFTs

I was struggling with a question, hoping someone could point me in the right direction. I'm interested in 2D CFTs on a cylinder. I want to take the tensor product of two CFTs. My questions are these:

(1) It seems that the total stress tensor will have modes that are the sum of the individual stress tensors $T^{(1)}+T^{(2)}$, like $T(z) = \sum z^{-n-2}(L_n^{(1)}+L_n^{(2)})$. I'm making a mistake in my calculation, but this operator should be in the conformal block of the identity, right? A descendant coming from applying an operator in the "full" algebra, like $(L_{-2}^{(1)} + L_{-2}^{(2)})|0\rangle$?

(2) I should get operators like $T^{(1)} - T^{(2)}$, and I can't see from where they would descend. I expect they should be primary operators, but I'm making an error I think. Is this a primary operator?

(1) The state $(L_{-2}^{(1)} + L_{-2}^{(2)})|0\rangle$ does correspond to the total stress tensor for the product CFT. Acting on this state with the lowering operator $L_{2}^{(1)} + L_{2}^{(2)}$ gives a state proportional to the vacuum (except in the case where the central charge $c=0-$then the stress tensor is primary.
(2) The state $T^{(1)} - T^{(2)}$ is not the state to consider; it's actually a linear combination of two states with well-defined (though different) properties under conformal transformations. The state built from $T^{(1)}$ and $T^{(2)}$ that has good transformation properties and is linearly independent with $T^{(1)} + T^{(2)}$ is the combination $(c^2 T^{(1)} - c^1 T^{(2)})|0\rangle$, where $c^i$ is the central charge of the $i^{th}$ CFT. Acting with Virasoro raising operators shows that this state is a primary state.