1) If a player move the gardener to a plot with 4-height bamboo does the adjacent irrigated plot still grow?
2) Gardener objective:
"If an improvement is depicted on the card, it means that this improvement has to be present on the plot in the playing area, in order for the objective to be fulfilled."
If on top of the depicted improvement an additional improvement was added to the plot, does the gardener objective still fulfill?
Here you're making a mistake.
You cannot place an improvement on a tile that already has an improvement.
To place an improvement the tile must be empty. Which means: no previous improvements AND no bamboo.