Changing voltage multiple times
What you're doing is certainly feasible, as long as you're willing to handle the waste power. The one change I'd suggest is replacing the PSU with a RAC04-12SC. You don't need the -12 that your specified PSU provides, and going to a single 12 volt output will giver you 333 mA, which may be enough. Note that, if the PSU is going full out, it will provide 4 watts, and the 7805 will have to dissipate 2.8 watts minus the power in the relay coil. This is certainly possible, but you must use a good heat sink, with proper care for thermal connection.