- washing babies (at some level you are washing off poo and wee)
Haven't personally, but would do if needed. It'd save my back doing it in the bath.
- washing pets (same as above)
My dog is far too big for the sink, but I do use the kitchen sink to clean out the cat litter tray, and hamster cage.
- brushing teeth (spitting in a food prep area)
I am expecting a plumber tomorrow, when they kids are getting ready for school. He'll be in the bathroom, so the kids' toothbrushes will be in the kitchen instead.
- emptying mop buckets (pour it down the toilet)
Don't use a mop bucket, but used the sink when I did.
- washing/dying/cutting hair in the kitchen- hair in food prep area
I cut my DC hair in the living room as it's easier to vacuum up. I did dye her hair in the kitchen a couple of weeks ago, as it'd be much easier cleaned from lino than carpet!
- washing clothes and shoes in a food prep area.
Nah, I use the mangle outside 
You do realise that the vast majority of people don't have a utility room? Some people even only have one bathroom, which isn't always accessible for certain tasks.