Zen Cart, static pages, custom error pages, custom index pages, show home directory tree.

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.