Let's call the embedding dimension of a manifold $e(M)$ as the minimum $n$ such that $M$ (smoothly) embeds into $\mathbb{R}^n$. Is it true that $e(M \times N) = e(M)+ e(N)$? I think the answer is "no", but I would like to see a counterexample. What about $e( S^2 \times S^2)$?
EDIT. They made me notiche that $S^1$ is a counterexample. The curiosity about $S^2$ remains, though. I feel like $e(S^n \times S^m) = e(S^n) + e(S^m) -1 = n+m+1$.
Sure, $e(S^n \times S^m) = n + m +1$. (Your formula is off because $e(S^n) = n+1$.) Think of the exact same construction you do with a torus: you fit $S^m$ as a small sphere in the normal direction to each point on $S^n$.
More precisely, $\Bbb R^{n+1} \setminus \{0\} \cong S^n \times \Bbb R$ by the diffeomorphism $x \mapsto (x/\|x\|, \|x\|)$. Thus $$\Bbb R^{n+m+1} \setminus \Bbb R^m \cong S^n \times \Bbb R^{m+1},$$ and of course this supports an embedding of $S^n \times S^m$.