Are hair appliances and cosmetics allowed in carry one in 2017
Any cosmetics that are liquids need to meet the relevant rules eg no single item more than 100ml, total to go in 1000ml clear plastic bag.
The application of the rules seem to vary slightly from airport to airport so, for example, I've been told to put lipstick in the bag at some airports and not others.
Yes although some cosmetics will fall within the definition of liquids and need to go into the bag. If the appliances are powered by gas cartridge there are restrictions on carrying spare cartridges as well.
Electrical hair styling appliances are fine, but make sure you check the voltage for wherever you're going to make sure they'll work when you get there.