I have two questions about formatting a definition in latex/beamer. Please run the short document below

This is not an answer (as I couldn't comment it) but rather to show you how to basically start using the beamer, frames and blocks. As @leandriis has mentioned you have to use begin{frame} to create a new slide and you may use frametitle to add a title for this slide . Also, the definition block should have a body and you can optionally place a title for it in a square brackets like this [Definition Term] and then you may italics it or not .

About the coloring you need to be more specific about what you want to color. Generally speaking you can use setbeamercolor command then adding the name of the block and the color of your choice of the background and foreground.

This is a minimal code (based on your example) you can start with:

documentclass[aspectratio = 1610]{beamer}
usepackage{amsthm}
usepackage{amsmath}
usetheme{Frankfurt}
usecolortheme{wolverine}
begin{document}
begin{frame}
frametitle{Slide title}
    begin{definition}[textit{Definition Term}]
        begin{enumerate}
            item How do I get this text to appear in standard italics ?
            item Also, is it possible to have this definition (with only a header) omit the white bar below?
        end{enumerate}
    end{definition}
end{frame}
end{document}

The output you get is