What material is the fur made out of?

According to the tag it’s 100% polyester and feels like polyester.

Does the top flap "biil" fold down to protect your eyes? If so is it held up by snaps or magnets

The Trapper hat bill is sewn in place with strong thread. If you could turn the bill down it would protect your eyes when you fall and hit the ground due to not being able to see. The Trapper hat is well made of quality materials and you'll be glad you have it when it's cold. Hope this helps.

Does this hat really have cats on it?

The black one does, can’t say for the other colors. They’re little tho and unless you look closely it just looks like lines and designs

Will this fit my 10 year old daughter and 7 year old son?

The trapper hat is 14 inches in length, 19 inch circumference. The face opening is 8 inches, and from front to back is 19 inches. The adjustable clasp is 5 inches . I think your 10 year old can wear it, but it may be to big for the 7 year old. Order the hats for them, and if it doesn't fit to your liking, just send it back. I hope they both can wear the hats, because I love mine.