Is $\mathbb R \times \mathbb R_{sorg}$ normal? I know that $\mathbb R \times \mathbb R$ is normal and $\mathbb R_{sorg} \times \mathbb R_{sorg}$ is not. But what about $\mathbb R \times \mathbb R_{sorg}$ ?
$\mathbb R_{sorg}$ is the Sorgenfrey line.

Yes, it is. By Dowker's theorem we know that $\mathbb{R_l} \times [0,1]$is normal  ( $\mathbb{R_l}$ is the left limit topology, another name for the Sorgenfrey line), because  $\mathbb{R_l}$ is generalised ordered, so normal and countably paracompact. 
Then a theorem by Morita (paper) shows that  $\mathbb{R_l} \times \mathbb{R}$ is also normal as the reals can be covered by countably many subspaces that have a normal product with  $\mathbb{R_l}$. 
[ADDED]
The above was my first idea. The Sorgenfrey line and the reals are so nice that more can be said:
The Sorgenfrey line is hereditarily Lindelöf (see this blog post), so in particular paracompact normal, and it's also perfectly normal. (items A, B,H in the above post)
Then result 2 in this post says that a product of a paracompact and a sigma-compact space is paracompact, and is proved in that blog as well. As $\mathbb{R}_l$ is paracompact and $\mathbb{R}$ is sigma-compact, we know the product is paracompact.
But result 4 in that same post gives even more: a hereditarily Lindelöf space (Sorgenfrey) times a separable metrisable space (the reals) is again hereditarily Lindelöf (which implies here that that is hereditarily normal, and even perfectly normal). The proofs here are simpler, because we can use more of the spaces than in the Morita and Dowker results.
