1. Can I reveal my objective in other player's turn? Maybe they make a land pattern or bamboo height I desired?
No. (We played that incorrectly our first game):
Takenoko rules p. 6 wrote:
At any time during his turn, a player may complete an objective if the conditions shown on the card are met.
This only matters for gardener objectives though. The plot pattern will not change once tiles are laid down, and you can still claim on your turn.
2. Can the panda eat bamboo on a plot occupied by the gardener?