Starting from the middle, Equius is not a joke character. I'll just directly quote the author commentary (page 2215) here.

"So one major challenge I found interesting was something like, "How do you introduce someone who appears to be almost instantly revolting or objectionable, and then proceed to make the reader love this person against all odds?" Equius is probably the strongest example up to this point of me embracing this idea as a challenge... There are many small ways his portrait is guided so that a guy who really has no right to be seen as endearing ends up being strangely so, often in ways that aren't even easy to explain. I firmly believe that if you don't love Equius after you've read most of Homestuck, then there's something wrong with you." -dclussie

Equius is a character struggling with an internal contradiction. He loves and upholds Alternia's brutal structures (even to his own detriment), but he's continuously proven wrong about them and kind of wants to be wrong. He ultimately fails to reject his upbringing, taking it to its natural conclusion at the hands of a clown. He ends up being the proof against his beliefs.

I hold that one of the single most important conversations in Homestuck is between Equius and Dave. Equius argues in favor of doing things the 'correct' way; the way that Dave is destined to play things out as the hero. Dave's rejection of this marks a milestone of personal growth. He no longer trusts authority figures or their weird fetishes, and chooses to do things his own way.