IC 4066 Switching on Relay
I think you are pushing the 4066 far beyond its limits by powering a relay coil from it. The chip heats up to a point that the other switches fail too ("current apparently bleeds over to SW2"). If the chip still functions after you manually 'reset' it by removing power to it, I think you are just plain lucky.

The solution will be that you power your relay with a separate transistor (and flyback diode). A bit like this answer