Short answer, it is a bug with the latest version of XML::SAX (specifically the XML::SAX::PurePerl parser module in that distribution):
You probably don't want to use the PurePerl parser anyway for performance reasons. The simplest fix is to install XML::SAX::Expat. That will then become the default SAX parser. Once you've done that, try the XML::Simple install again. The errors should be gone.
No, running the previous version of XML::SAX is unlikely to cause a problem although parsing of XML will be very slow.
The error you got when trying to install XML::Parser was simply because you don't have the expat header files installed. Typically the expat library would be in a package called something like libexpat and the headers would be in a package called libexpat-dev. You'd need both to build XML::Parser.
On Linux servers, it's generally easier to install the OS packages for CPAN modules if they're available.
No, don't uninstall XML::SAX, it provides the base class used by all the SAX parsers, so it's a prerequisite for XML::SAX::Expat.
In the beginning, there was only XML::Parser. It provides a good fast parser (via libexpat) but it's API isn't very friendly. Then the SAX API was created to serve as a common API. This means that you can plug in different parsers without changing your code. The last SAX parser you install will be the system default from that point on. The Perl XML FAQ has links to Kip Hampton's introductory articles.