First off, I'm glad you like what you see! I hope it's as fun as it sounds.
Your questions are interesting. I'll take a crack at them.
1) I didn't provide much about SBURB as a game because the sessions we follow in-story are all described as outliers. That said, if I wanted to hew a little more closely to what we're led to believe is a "traditional" session of SBURB, I'd probably steal the idea of Progress Clocks from games like Blades in the Dark. For instance, I might progress a "Death of the White King" clock at certain story points, or when players fail to avert certain forms of trouble. Similarly, I might break the process of entering the session into a clock based on its key steps (i.e. Install Sburb, Deploy Devices, Alchemize Entry Item, and Prototype Sprite [once or twice]), then allow the actual entry to proceed at the pace that drama demands. The process of building up a tower and sufficient grist to help create a Universe Frog would also be a clock (or maybe one clock per player). That sort of thing.
All that said, if your players are familiar with Homestuck already, much of that stuff could be glossed over in favor of them chasing shenanigans in the course of broadly-stated goals.
2) The GM Principles should guide you in when to use your moves, but in general "Threaten trouble, individually or massively," "Impose an obstacle," "Hint at secrets," "Take something precious," and "Offer something the players want or need" are the moves you'll use to prompt players to act. "Follow through on a threat," "Reveal the truth," "Mark a connection for a player," and "Advance the clock" are moves you'll use to acknowledge/reward/respond to player actions. ("Mark a connection for a player" could use more clarity--it's intended to let the GM give out free connections to encourage cool stuff and power up players in ways that help the story). The remaining GM Moves are more focused on scene-setting and plot progression/expansion, and are good for when the current scene is more player-driven or has nothing in particular going on.
Of course, if you use Progress Clocks as suggested above, the timing for a lot of GM Moves would become more obvious. In fact, it'd be fair to add a GM Move of "Set a Progress Clock," or maybe something more SBURB-appropriate like "Reveal a Countdown."
Hope these help!