The game lets me build the beauty parlor, but when it's finished it doesn't work. It activates the tutorial for it, but there is no option to apply tattoos on characters. The upgrade menu shows the build meter as full, but the project disappears from the upgrade list like actually completed ones.

When you are in the equipment menu at the bottom right there is a Button.

This button leads to the tattoo slots. Just equip an ink there.

There isn't.

Do you have inks in your inventory? They are made with Alchemy. I don't remember ever seeing them on sale, although the base ink is.