Both questions have been asked before. The answer to Question 1 is negative, see [here][1].
Question 2 appeared just some hours ago [here][2]. 


  [1]: http://mathoverflow.net/questions/36126/on-the-size-of-balls-in-cayley-graphs/36132#36132
  [2]: http://mathoverflow.net/questions/43964/folner-sequences-of-amenable-groups-of-exponential-growth