Spin groups and the Quaternionic Representation

Question

Quaternionic representation is also known as the pseudo real representation.

For the $Spin(n)$ group, it looks that only $Spin(3+8k), Spin(4+8k), Spin(5+8k)$ representation are pseudo real.

Questions:

1. Let us accept that the group isomorphism shows that $$Spin(4)=SU(2)\times SU(2)=Spin(3)\times Spin(3).$$ Let us accept that we know $Spin(3)$ representation is pseudo real.

Then Don't we have the $Spin(3)\times Spin(3)$ is in the real representation (pseudo real $\otimes$ pseudo real can make a real representation (!?)), thus the $Spin(4)$ is also in the real representation?

But according to the above, the $Spin(4)$ is also in the pseudo real representation?

2. We know that $Spin(n)/\mathbb{Z}_2=SO(n)$, since $Spin(3), Spin(4), Spin(5)$ representations are pseudo real representations, what can we say about the corresponding representations of $SO(3), SO(4), SO(5)$ --- can they also have a pseudo real representation?

Thanks for the answer in advance!