I've never come across hair removal in my research of the Tudor era so I just don't know. A dislike of body hair is quite a modern thing in that I remember going to various countries as a teenager and being shocked by women with underarm hair and hairy legs. I'm not sure that Tudor women would have worried about it.
As far as cosmetics are concerned, there is a great fact sheet at http://www.npg.org.uk/assets/files/pdf/exploring-elizabeth/npg_tudor_makeup.pdf on what kind of make-up the Elizabethans used. For a white facepaint they'd mix egg whites, powdered egg shells, Alum, Borax and White poppy seeds. Lips and cheeks were reddened with a mixture of Vermillion (red crystalline mercuric sulphite), Gum Arabic, Egg white and Milk from figs, or a lipsalve made from cochineal. The fact sheet also suggests that Elizabeth dyed her hair red with a mixture of lye, wood ash and water. Nice!
I remember reading about Cleopatra days they used even stones to remove hair. Sounds painful but I’ve read it more than once. Now that might explain why Anne of Cleves was such a turn off. Don’t bash me since my ancestors on my Dad’s side came from Germany but maybe she didn’t practice the same habits. His dislike seemed to be instant even before greeting her.