How can I have a rising edge close a relay for a fixed time?
The solution is called a monostable multivibrator, or MMV. When it detects a rising (or falling) edge it gives a pulse at the output with a chosen duration. You use that pulse to trigger the relay. This is a typical job for an LM555 timer.  

If the signal to be switched is DC you can use an optocoupler, like Matt suggested. A CNY17 may be a good choice. If the signal is AC you could use an SSR like the LH1500, or a reed relay. I wouldn't use a power relay because they're not made to switch low current signals.