Add 1x2's to the stud then 3/4" drywall that should be 1.5"

No link has been proven exactly between black mold and health issues. It most likely does cause issues when the exposure is high enough but as a health adult you should be fine to clean up the mess.

Wood should be fine if iwas only wet for a day.

Mold should be fine for you unless you have health issues, wouldn't put your child around it though.

I'd cut the drywall out as far as it got wet, get some fans to dry everything out then replace the drywall cut out.