I have a Zabbix 2.0.5 instance.
If I configure a web test in a host with monitoring configured to go through a Zabbix proxy, the test won't ever run.
When I change the host settings not to use a proxy the test runs again.
The question is: can I run web checks from a different host than the main Zabbix server? How?
You can since Zabbix 2.0: https://www.zabbix.com/documentation/2.0/manual/introduction/whatsnew200#web_monitoring_by_zabbix_proxies
Make sure proxy is compiled with web monitoring support (--with-libcurl parameter - check proxy startup messages in the logfile).
Remember that proxies update their configuration data once an hour by default. If it is an active proxy, you can try restarting it. If it is a passive proxy, for testing set ProxyConfigFrequency in the server configuration file to 300 or so and restart server.