"The Dog" is Apollo, the player character of the game the protagonist is playing. The game is a very difficult precision platformer, and thus Apollo "dies" via collisions with enemies, spikes, etc. dozens if not hundreds of times per episode. That said, Apollo is stated to be immortal and incapable of truly dying, and every "death" is just Apollo turning into a white light, disappearing, and respawning harmlessly back at the last checkpoint. (MAJOR exception: In Act 1 episodes 8 and 9, Apollo doesn't die but neither does he respawn. Instead, he remains impaled on a bed of spikes, alive but in severe pain, until he frees himself.)
While not shown on-screen, dialogue from flashbacks and memories heavily implies that the protagonist was bullied in school and discouraged away from his interests and passions by his unsupportive parents.
Though not shown on-screen, a character recollects about and describes a time in the protagonist's youth in which bullies stole and destroyed his Sonic the Hedgehog doll.
Apollo is a character in a video game that the Will is playing, and thus is subject to being controlled by Will. In Act 2 episode 3, Will has Apollo dive down and remain underwater, almost drowning him but relenting at the last possible moment. Apollo is shown to be in clear distress over this.
Never directly stated or explained, deliberately left ambiguous and open to interpretation, but the "Status" message in the video game often APPEARS to mirror the player's real-life mood or circumstances. Likewise, the player character, Apollo, APPEARS to be intelligent or "real" in some unspecified way.
Not shown onscreen, but dialogue from this episode recalls and discusses a time in the protagonist's childhood that heavily implies that a classmate of his killed himself.
A video game character is constantly having to struggle with navigating water levels and swimming sections on a limited air supply. This is a recurring theme throughout the series.