The first clip (zip from cork) will always work if RIGHT is released at least 2 frames before jumping, the jump starts one frame before hitting the wall, and JUMP is held for a duration of 1 frame. RIGHT only needs to be depressed long enough to zip. RIGHT can be held 2 frames longer if the initial x-subpixel is 0-99 (39% chance). Holding JUMP for a duration of 2 or more frames will work if the initial y-subpixel is 128-255 (50% chance). When Sonic clips into the wall, the y-subpixel will remain unchanged with a 1-frame duration jump and will change by 128 with a 2+ frame duration jump.
The following information regarding the second clip through the waterslide refers to the starting position werster had when manipulating the pixel position and ignores that a jump inside the wall happened (the first jump inside the glitchy looking wall). The x-subpixel must be 0-46 and the y-subpixel must be 128-255. RIGHT must be held for 4-5 frames and JUMP must be depressed on the 5th frame for 2 frames. ( [>][>][>][>][>A][A] or [>][>][>][>][A][A]). This can also be performed with shorter RIGHT presses as long as JUMP is pressed on or one frame after the 4th frame that RIGHT is held. The x-subpixels can be manipulated to be within that range by jumping from the correct x-pixel and tapping RIGHT at a time where Sonic is against a wall (only the second tile from the top is left solid (16 pixels)). A tap of 3 frames or less will work 100% of the time.
The y-subpixel can be manipulated, but not with 100% accuracy, by performing a jump with a duration of 'x' frames. For reference, the following list shows the odds of x-frame jump landing with good y-subpixels, split into two groups in this format (x-frame jump, chance of good subpixel): [initial y-subpixels 0-127] (1,12.5%) (2,37.5%) (3,75%) (4,75%) (5,50%) (6,62.5%) (7,75%) (8,62.5%) (9,87.5%) (10,62.5%) (11,12.5%) (12,25%) (13+,62.5%) [initial y-subpixels 128-255] (1,87.5%) (2,37.5%) (3,62.5%) (4,25%) (5,50%) (6,12.5%) (7,62.5%) (8,62.5%) (9,75%) (10,37.5%) (11,87.5%) (12,75%) (13+,37.5%). The 0-127 group has better or equal chances of success for jump lengths 2-10 and 13+ frames with better chances with 3,4,6,7,9,10,13+. 0-127 will always be the initial y-subpixel with a 2+ frame JUMP press when performing the first clip.
So, to summarize a way to manipulate the y-subpixel for the second clip, perform the following: Hold JUMP for 2+ frames when performing the first clip (forces y-subpixel to 0-127), zip to the correct x-pixel where the second wall clip's zip will start (if Sonic zips past this location, zip back to the right or left as needed, do not jump to get there), hold JUMP for 3-10 frames, when Sonic reaches the peak height of the jump tap RIGHT for 3 frames or less (a few extra frames still gives a good chance of a good x-subpixel).
The third zip with works with holding right all the way to the level wrap with any set of subpixels and, additionally, Sonic will be pushed to the correct x-pixel even if Sonic falls a few pixels from the left-most position against the wall.
The following is just extra information, explaining why either clip failed. When performing the first clip, if Sonic falls down in the wall, the jump either started a frame early, or had bad y-subpixels with a 2+ frame jump duration. When performing the second clip, if Sonic jumps straight horizontal, jumps horizontal except pushed down one block, or jumps straight vertical, the x-subpixel was very high (232-255). In all other cases, Sonic should jump with a slight up angle and gets pushed out if subpixels were bad, RIGHT was held too long, or JUMP was not held for 2 frames.