If you haven't already, you might first build a phpinfo() page to check the resource (memory) limit and the maximum execution time. Having either one set too low can cause the final step to fail.
If you put your rewrite rules in the apache config file instead of using .htaccess, copy the new rewrite rules in .htaccess_root to your apache config file. Otherwise, at various points in the install process you'll end up with a blank page. At least this was happening to me on 3.10 installations.
I had a very similar issues perviously (see http://ez.no/developer/forum/install_configuration/ez_4_0_0alpha2_setup_wizard_jumps_back_to_first_page ). In my case it was the Suhosin php-extension that was preventing the ez-installer to download the site-packages. What Suhosin does, among other things, is to disable url_fopen. So I think this is the reason in your case too, since your hosting provider has disabled allow_url_fopen. The eZ 4 installer seems to behave different than in previous versions.
So I think the only solution is to either aks you provider to allow url_fopen temporarly during installation or to manually install ez and import the plain-site (or whatever package) manually.
It's a me too comment: EzPublish did not finish the last step and I suppose that my Web provider might not be generous enough... Here is what I've got:
32 MB RAM max
10 s execution time per script max 24 processes
Please, could anybody report the minimum requirements for EZ 4.0? Not everyone is running his/her own Web server but relies on a provider. In my case the values above already represent the 'business' version - there is no larger package but a server... Thus it would be great if the data are good enough...Thanks!