"Free" Hopf algebra

1) Yes, free means free as an algebra. (I don't think it could possibly mean anything else. "Free Hopf algebras" and "free coalgebras" as left adjoints to the forgetful functor don't exist.)

2) is elementary, I think. Let $H$ be a connected free monogenic Hopf algebra. Let $x$ be a generator. Thanks to the connectivity assumption, $\deg x > 0$. The algebra $H$ is the free graded-commutative algebra on $x$, either $\Bbbk[x]$ if $\deg x$ is even or $p = 2$, or $\Lambda(x)$ otherwise. Then by counitality and degree arguments, $\Delta(x) = 1 \otimes x + x \otimes 1$, so $H$ is indeed either a polynomial or exterior Hopf algebra.

Note that I used the connectivity assumption above. If you don't assume that $H$ is connected (or at the very least locally conilpotent) then the result becomes false. Consider for example the Hopf algebra of dimension $2$, with basis $(1,x)$, both in degree zero, $x \cdot x = 1$, $\epsilon(x) = 1$ and $\Delta(x) = x \otimes x$. In other words, the group algebra of $\mathbb{Z}/2\mathbb{Z}$.