There is a powerup that squashes Anton's or Annie's whole body into a ball that looks like their head. Danton in one level can get his whole body squashed by a tree, though it heavily emphasizes his head in cartoonish fashion.
The final boss has a phase that vaguely resembles arachnids, though it's brief and you don't fight it directly. Then again you do run from him during this phase, and that might be scarier.
Getting a certain power up squishes Anton or Annie into a ball and it looks claustrophobic. Other than that some areas can feel more claustrophobic than others, but claustrophobia isn't emphasized.
Anton is obsessed with hard alcohol, to the extent that the goal of the game at large is to recollect his stolen spirits. This likely contributes to his rude and aggressive demeanor.
Satan mocks the player with a "boo-hoo" and a mock crying face for presumably being sad that the game is almost over, but obviously the player's gender is irrelevant here.
Sort of? Satan is dissatisfied with not being red enough after discovering there is someone redder than he is. Other than this, he seems quite proud of his appearance.
Satan's sentient mirror takes a hit and is seemingly destroyed, although it's been revealed that they survived, and it's unclear whether they meant to sacrifice themselves or not.
Huge explosion that levels an entire city as well as presumably all of Hell. For what it's worth, all of the main characters apparently make it out unscathed.
Religion isn't discussed in a philosophical way, but religious imagery frequently crops up, such as the obvious depictions of Hell/demons and a boss based on Biblically accurate angels.
Anton and especially Annie could be read as mentally ill and are extremely violent, though this is unconfirmed and could largely be a result of alcohol and caffeine abuse respectively. It's a (playable) cartoon so the characters' personalities are far from realistic.
Most characters' sexualities are not touched upon. There are only two canon relationships and they're both heterosexual. Satan is vaguely implied to be bisexual but he's implied to survive anyways.