Continuous function between a lower semi-continuous function and an upper semi-continuous function.

Yes. See Katětov–Tong insertion theorem. I suspect this result for your context (compact metric space domain) was proved earlier, but I off-hand I don't know. FYI, googling semicontinuous + insertion + theorem will lead to many similar results. (a few minutes later) I just noticed you have strict inequality, whereas the theorem I cite involves non-strict inequality. I'm not sure whether your version can be obtained from what I gave, at least unless we assume (everywhere locally) a positive lower bound on the pointwise differences between the semicontinuous functions.

It is true. See the book

Engelking, Ryszard. "General topology."

On p.428 5.5.20 you find the following result as an exercise:

A $T_1$-space $X$ is normal and countably paracompact if and only if for each pair $f,g$ of real-valued functions on $X$, where $f$ is upper semicontinuous and $g$ is lower semicontinuous such that $f(x) < g(x)$ for all $x \in X$, there exists a continuous $h : X \to \mathbb R$ such that $f(x) < h(x) < g(x)$ for all $x$.

You will also find references to papers containing proofs, for example

Miroslav Katětov, On real-valued functions in topological spaces, Fundamenta Mathematicae 38 (1951), 85–91

as quoted in Dave L. Renfro's link.