For the bath mat, I would lightly scrub into the back with baking soda and rinse with vinegar. Do again. You can also just soak it in a your washer in a vinegar baking soda combo for an hour and then wash with your normal detergent and a 1/2 cup of baking soda.
Dry erase marker is one of the hardest things to remove...ugh.
I always try vinegar baking soda first it works with most everything, but if i remember corrrectly it does not on dry erase. I ended up using Goo Gone and Goof off. I scrubbed in that, then rinsed with room temp water, then rinsed in a vinegar solution of 1 cup to 1 gallon water, then rinse again in plain water. This will break down the chemical residues so your kids won't have reactions to them.
You can buy the 5lb bag of baking soda and two gallon box of vinegar from costco or sams. Its about 8 dollars for the combo and it last me about six months, and I used it on most everything to clean, including in my laundry. CHEAP < CHEAP but works awesome.
My cousin called the other day with a complaint that her cat had lost its mind and peed all over her leather couch and the urine had soaked into the foam cushions and in her porta crib while they we gone. She had tried just about everything expensive. She tried the bs and vin combo and it worked!! No more urine or smell. It actually changes the PH of the urine and breaks it down ...Works on red food dye stains in carpet and clothes too...but anyhow, sorry for rambling, I just love cleaning and for cheap so I hope this helps!!
K.