All spaces are separable metrizable.
My first question will be about the Borel class $F_\sigma$, but I really want to know if they generalize for all Borel types ($F_\sigma$, $G_\delta$, $F_{\sigma\delta}$, $G_{\delta\sigma}$, etc.)
1. Does "$X$ is an absolute $F_{\sigma}$-space" usually mean the same thing as "$X$ is an $F_{\sigma}$-space"? And does this mean $X$ is an $F_\sigma$-subset of every metric space in which it is embedded? If so, then $F_\sigma$-spaces would be precisely the $\sigma$-compact spaces.
2. What is the definition of "$X$ is an essential $F_{\sigma\delta}$-space"? Does this mean $X$ is an $F_{\sigma\delta}$-space but does not belong to any lower class in the hierarchy, e.g. $X$ is neither an $F_\sigma$-space nor a $G_\delta$-space?
A space $X$ is an absolute Borel set of class (type) $K$ (e.g. $F_\sigma$) iff whenever we embed $X$ into a metric space $M$ via a homeomorphic embedding $i :X \to M$, $i[X]$ is of that class $K$ in $M$. This notion is due to Kuratowksi, see this more recent paper for a reference.
an absolutely closed set is compact, as is easily seen, and an absolute $F_\sigma$ is a $\sigma$-compact space.
an absolute $G_\delta$ is completely metrisable (also a classic fact).
Absolute $F_{\sigma\delta}$ and $G_{\delta\sigma}$ spaces are more complex, e.g. see the aforementioned paper e.g. , also see this paper by Stone.
As to the "essential" part, your hypothesis on its meaning (of that absolute class but no simpler one) makes sense; I could not find an authorative reference for it, though.