Takenoko: What happens if you draw a card that has an existing pattern in the garden?
You draw a plot card and that exact same shape is already existing in the garden
  (including the right parameters) Can you just show your objective card as fulfilled and
  get the points without doing anything else?


Yes. At any point in your turn, you can play as many fulfilled objective cards from your hand as you like. I've found this often happens at the end of the game... drawing a card in hopes that the objective is already fulfilled is a decent strategy.


  You draw a gardener card and you need 3 four-level bamboo in the garden. If there's an 
  existing four-level bamboo in the garden, do you only need to do 2 more?


Yes, provided they're the same color. As it often happens with objective cards, other players may be configuring the garden to suit your objectives without even knowing it.