Does a pair of sixes beat a pair of aces?

No.

In case of a tie, the pair of the highest rank wins. If ranks are tied, then the hand with the highest non-paired card wins. If that card is also tied, then the hand with the next-highest non-paired card wins, and so on.

Examples:

Hand 1: AAK42, Hand 2: 66543. Hand 1 wins, since A is higher rank than 6.

Hand 1: KKQ42, Hand 2: KKJ98. Both players have a pair of Ks. Hand 1 wins, since Q is higher rank than J.

Hand 1: KKQ42, Hand 2: KKQJ2. Both players have a pair of Ks, Both players have a Q. Hand 2 wins since J is higher rank than 4.