$-1$, sum of squares and embedding in $\mathbb{R}$

Question

In the proof of theorem 1.4, Algebraic K-theory and Quadratic forms, Milnor uses the following fact:

If -1 is not a sum of squares, then F can be embedded in a real closed field.

I thought of some examples and "counter-examples" of this proposition. For all finite fields $\mathbb{F}_p$, $-1$ is sum of $p-1$ squares (and this is also true for all field of characteristics $p$). All real algebraic extension of $\mathbb{Q}$ should satisfy this proposition too (since there exists $\mathbb{Q}\hookrightarrow\mathbb{R}\hookrightarrow\mathbb{C}$). $\mathbb{Q}(t)$ can be embedded in $\mathbb{R}$ by sending $t$ to any transcendental element of $\mathbb{R}$. It's likely that $\mathbb{Q}(T_1,...,T_n)$ can also be embedded in $\mathbb{R}$ by the same method.

Question (I'm sorry for multiplicity):

1. Is $-1$ a sum of squares in the field of formal series $\mathbb{Q}((t))$?
2. Can $\mathbb{Q}((t))$ be embedded in $\mathbb{R}$?
3. What are some real closed field rather than $\mathbb{R}$ and $\overline{\mathbb{Q}}\cap\mathbb{R}$?
4. What might be a proof of the above fact?