That hole sounds like where the hazard lights button used to be, yes.
The high beams should burn at low intensity when the car is running, that's the daytime running lamps (DRL), but not the low beams or parking lamps...if it's all on during the day, something may be blocking the light to the dash sensor that tells it to turn the lights on when it gets dark (if equipped with this option)...the sensor is on the driver's side of the defrost vent up by the windshield...it's a little hole with the "eye" in it. Sometimes crud gets in there and blocks the light, and it thinks it's perpetual night....
The high/low beams are activated with the turn signal stalk, as well as the wiper functions, cruise control, and that emergency flasher is tied into it too....you'd probably have to replace that whole assembly to have a new flasher button...they are quite expensive (called multi-function switch) and require steering wheel and airbag removal to replace...
Where is the other button located? I don't have a trunk button, but I assume it's in the glovebox? I'm drawing a blank on this one...