Deepwood Shrine is the first dungeon in the game and therefore one of the easier ones.
The dungeon mainly consists of rolling through enemies correctly and optimizing movement.
Rolling upwards after entering the room makes it possible to combo one of the three Mulldozers against the wall, allowing a quick kill on it. (Video Guide at 2:45)
After getting the lillypad small key and returning to the previous screen, the pot on the switch will rarely not spawn on the switch. This can be avoided, as seen here.
Standing on the side Chuchu leans first ensures he only sways 3 times, standing on the opposite side will make him sway 4 times and lose time.
Keep Chuchu in the middle of the room so when you suck up all his goop he does a fast sway or "half sway".
On the third phase, he will either jump once then rush you, or jump 5 times. Jumping once is preferable, but it is up to RNG and the general strategy doesn't change.