Fuel Pump
#1
#2
What you hear is not the relay, that would be a very soft click if you could hear it at all; you hear the pump pressurizing the system. If you're not getting fuel out of the filter there is blockage somewhere in the fuel line between the pump and the filter outlet. The most obvious possibility is the filter although it's unusual for one to suddenly clog completely. Filters are cheap enough and need to be changed occasionally anyhow. I'd pull the filter and with it removed turn the key on for a couple of seconds to see if the fuel system will pee in a bucket. If it does simply put in a new filter and you're done. If not, work your way upstream till you find the problem.
#3