DRL questions

Hi everyone,
I just recently got a Blizzard Pearl 2013 Persona. I have a few questions about the daytime running lights.

First of all, I've noticed something odd. If I turn on the car, the DRLs come on just fine, but if I switch to the headlights, then switch back to just DRLs, they no longer come on. However, if turn the switch to the "off" position and wait a few minutes then turn it back to "DRL," they come on again. It's as if turning on the headlights sets a timer before the DRLs can come on again. Is this normal? Or is there something wrong with my system? (Note: I have 5000K HIDs installed).

Second, as far as I know there is no simple way to have the DRLs on when the headlights are on (sort of like pseudo-fog-lights). I've seen the different mods where you can wire the DRLs to the inverter/converter fuse and other similar things, but then they would be on whenever the car is on (in other words the "off" position on the headlight switch means nothing). Has anyone been able to locate a relay for the low-beam headlights? I know where the DRL relay is, and if there is a low-beam relay then I can just connect the two, problem solved. My other option is to just wire the main headlight connector to the DRL connector (positive-to-positive) on each side. Any input or suggestions?

Sorry for the long post. All input is welcome. Thanks everyone!