Here are the answers to your questions.
Do you support Zen Cart?
Yes, Standard plan & above
Do you support pseudo static pages, particularly URL rewrite?
Yes, you have complete .htacess file control
Do you support custom error page for http 400, 500, etc.?
Yes, you have complete .htacess file control
Do you support the customization of index pages?
Yes
Is it possible not to show the home directory tree if no index page can
be found?
Yes
Do you support ZendOptimizer?
Yes, we have better RoundCube installed
Can I upload the zip files, such as .RAR, .ZIP, to the ftp server, and
unzip files on the server?
Yes, our File Manager has Compress / UnCompress functions for ZIP, TAR
files
If you are thinking about implementing an eCommerce operation, I would
suggest our Professional Plan or above.