This guide will assume that you have a decent-enough understanding of Vivian Clark (and Soda Drinker Pro by extension) to have gone looking for this guide in the first place.

"Save your friend using dynamite" and "Explore as a fish" are two objectives in Vivian Clark that are located in the same level.

This level (henceforth referred to as "the fish level") is only accessible from the location shown in the "Explore as a fish" frame, a 3D platformer level. If you're having trouble locating the 3D platformer level, it can be reached via a painting in the skyscraper location shown in the "Present 200 collects" frame.

The fish level is a little annoying. It's so large that it takes several minutes to explore from end-to-end, and it's notably buggier than any other level in the game. For starters, the seafloor appears to be drawn in the wrong location; though it looks like the bottom of the level, you can swim underneath it, and you are actually supposed to do so to complete the "Save your friend using dynamite" objective. Additionally, the enemies in this level - long, orange fish with jagged teeth - cannot harm you whatsoever. On top of this, completing the "Explore as a fish" objective sends you to the wrong location; instead of sending you to the "head level" shown, it sends you to the "blue worm" level that is meant to be accessed from another character located directly below the starting position in the fish level.

Most seriously, the level contains a potential softlock that can prevent you from completing Vivian Clark. If you happen to complete "Explore as a fish" before completing "Save your friend using dynamite", then both the dynamite and the friend will despawn, and you will never be able to complete "Save your friend using dynamite" (and, thus, Vivian Clark) on the current save file. Your only option is to start a new save file and ensure that you complete "Save your friend using dynamite" before you complete "Explore as a fish".

Assuming you have not locked yourself out of "Save your friend using dynamite", it can be completed as follows:

(1) Enter the fish level from the 3D platformer level. Something you may not have known that I wanted to include as a cute little note: once you're in the fish level, you can return to the 3D platformer level if you immediately swim into your previous character, who is now sinking into the water; the springboard on the fish's back will launch them back up to the surface. Of course, it goes without saying that you don't need to do this if you're trying to save your friend using dynamite.
(2) As the fish, from your starting position, swim directly downwards, through the seafloor.
(3) As soon as you are through the ground, swim precisely down and to the left (if you are using a keyboard, hold A and S).
(4) While you are swimming down and to the left, look out for a green patch of ground that will be visible in the top-left corner of your screen. Keep swimming down and to the left.
(5) After passing the green patch of ground described in (4), keep going down and to the left until you see a collect. When you see this collect, immediately start swimming down, following the trail of collects to the bottom of the level. The dynamite is located at the very bottom, to the left of the last collect.
(6) After collecting the dynamite, swim directly up, maneuvering around any green clumps of ground you come across - keep trying to swim directly up. Eventually, you will come across an opening that will allow you to swim back above the seafloor.
(7) Once you are back above the seafloor again, go directly to the right, where the friend is located. Swim into the barricade while holding the dynamite to save your friend and complete the objective.

Nobody knows. The developers have been aware of the bug for several years but haven't been able to figure out the root cause.

Across Vivian Clark, holdable items are generally replaced with a red collect (worth 5 collects) when their associated objective is complete; for example, the letter in the crocodile-dodging-meteors stage, the yo-yo in the spiky-haired boy level, and the blue key in the blue boy level are all replaced with red collects once delivered to the proper places. This appears to be the case for "Save your friend using dynamite" as well; if the objective is completed, and if the fish level is exited and re-entered without completing "Explore as a fish", then there will be a red collect next to the friend you freed (which is a little unusual, as the red collect would normally replace the item itself i.e. the dynamite in other cases).

Leaving the level and returning after completing "Save your friend using dynamite" without completing "Explore as a fish" also causes the level to act strange; upon entry to the level, the game plays a sound effect as though you have picked up an item, though you will not have actually picked up any items.

For some reason, completing "Explore as a fish" completely removes everything related to "Save your friend using dynamite" from the game regardless of whether or not "Save your friend using dynamite" has been completed. The dynamite, the barricade trapping your friend, your friend, and/or the red collect (as applicable) will all be gone entirely, with no way to bring them back. The red collect and friend are still there in my completed save file, though. I have no idea.