After the first cutscene it is possible to break the box in front of the digging hole earlier than intended. This saves 2 frames compared to breaking the box after the cutscene starts.
1) Dash after the first cutscene
2) Immediately dash straight forward
2) Hold up left and count Wolf Links steps up to 6
3) Right after the 7th step, mash b for 3 attacks.
4) After the third B-attack hold left and B-attack again.
After entering the staircase it is possible to skip the Midna Text by jumping to the other side of the stairs in a specific angle. Note that if you jump too far to the left you will jump over the trigger and therefore have to do it again.
Upon entering the Rooftops of Hyrule Castle, there is a 1 frame window to skip the Midna text by jump attacking right after skipping the cutscene. You can use Input storage to constantly get it by holding L+A while skipping the cutscene.