Yes, I would take a gift, you would want her friend to bring your child a present. You can just get a cheap games/toys from the dollar store.
Yes, it is appropriate to take a gift.
Gift cards are not appropriate for that age child. Definitely take a gift to any birthday party. If it's just a playday .. no gift needed. You will be able to find a gift for same or less money even at a drug store. Also, only her food is provided from the hostess. Any other family members are just there for guidance and transportation needs.