I am new to geometric group theory and I am trying to read a bit to expand my horizons. I have encountered the following theorem: Suppose that $G$ is a group that has a free action by isometries on $\mathbb{R}^n$ with the Euclidean metric. Then $G$ is torsion free. I am wondering about generalisations of it:

I think it follows from the proof that if an isometry (that is it the cyclic group generated by it) acts on $\mathbb{R}^n$ and it has a finite orbit, then it has a fixed point. Am I right?

I saw somewhere that the theorem holds if one can replace isometry by a continuous function. I think I can prove it for $n=1$ because then a convex set is the same as a connected set, namely, an interval. Does anyone know a reference for the proof?

I assume the theorem fails if $\mathbb{R}$ is replaced by $\mathbb{Q}$. Can anyone give a counter example?

What happens if $\mathbb{R}$ is replaced by the $p$-adics?

Is there a version in characteristic $p$, but maybe instead of torsion free you cannot have orders that are coprime to $p$?