Existence of a function between two functions $f$ and $g$ where $f \in o(g)$
Yes.
For example, let $h=\sqrt{fg}$, assuming $f$ and $g$ are positive.
If not, we can let $h(x)=\max(\sqrt{|f(x)g(x)|}, g(x)/(|x|+1))$. The term, $g(x)/(|x|+1)$ makes sure that $h(x)$ will stay positive eventually.

Exercise. For each two functions $f$ and $g$, where $f \in \omega(g)$, there exists a function $h$ where $f \in \omega(h)$ and $h \in \omega(g)$.