I'd like to have the questions numbered, and for the first question of each chapter to be numbered one. I'd also like the title of each question to be left justified and inlined with the text
I suggest you set up a dedicated enueration list (called qq below), using the machinery of the enumitem package. For "smart" cross-references to questions, I suggest using the cleveref package.

\documentclass{amsbook}
\usepackage{lipsum} % filler text
\usepackage{enumitem}
\newlist{qq}{enumerate}{1}
\setlist[qq]{label=\arabic*.,ref=\arabic*,wide=0pt}
\usepackage{cleveref} % load this package *last*
\crefname{qqi}{question}{questions} % text labels associated with question items
\begin{document}
\chapter{First}
\section{Hello}
\lipsum[2] % filler text
Cross-references to \cref{qq:bb,qq:dd}.
\section*{Questions}
\begin{qq}
\item Title of question 1. Body of question 1.
\item \label{qq:bb}Title of question 2. Body of question 2.
\item Title of question 3. Body of question 3.
\item \label{qq:dd} Title of question 4. Body of question 4.
\item Title of question 5. Body of question 5.
\end{qq}
\end{document}
\subsection*{...}directive, followed by a list of problems, at the end of each chapter. – Mico May 14 '18 at 03:17xcbinamsbookthat is meant for this situation. please read the user documentation, which is the "ams author handbook" for monographs. available on ctan, or from a link on the page https://www.ams.org/authors/author-handbook – barbara beeton May 14 '18 at 12:18