Vertical Spacing Before and After "Cases" Environment

That's the normal \abovedisplayskip used by amsart, which is about 41% of the value used in the standard classes, 4.9pt against 12pt. The difference is really noticeable, but it's the AMS house style.

If you deem the value too small, change it: add

\AtBeginDocument{%
 \setlength{\abovedisplayskip}{12pt plus 3pt minus 7pt}%
  \setlength{\abovedisplayshortskip}{0pt plus 3pt}%
  \setlength{\belowdisplayshortskip}{6.5pt plus 3.5pt minus 3pt}%
  \setlength{\belowdisplayskip}{\abovedisplayskip}%
}

before \begin{document}. If you don't plan to typeset math in different font sizes, this should be sufficient.

In my opinion, all those long equations should be left aligned: the big chunks of white space on the left makes for difficult reading. The same for the cases environment.