Learn Swedish: Knappt syns kometen i teleskopet. - The comet is barely visible in the telescope.

The restrictive adverb knappt (barely) is fronted for rhetorical emphasis to stress how faint the comet is. Because Swedish enforces the V2 (verb-second) rule, moving the adverb to the front forces the verb syns into the second position, swapping places with the subject kometen.

How would this sentence be structured in a more standard, everyday context?

In normal word order, you would start with the subject and put the adverb in its usual mid-position: Kometen syns knappt i teleskopet. Fronting knappt makes the sentence sound more dramatic, focused, or literary.

Why use the s-form syns here instead of a passive like kan ses (can be seen)?

While kan ses is grammatically fine, syns functions here as an active, deponent verb meaning 'is visible' or 'shows'. It is the most natural and common way in Swedish to describe whether something can be perceived by the eye, rather than focusing on the action of someone looking at it.

Does fronting a restrictive adverb like knappt trigger any special grammar rules besides V2 inversion?

No, standard V2 inversion is all you need. Unlike English, which sometimes requires extra auxiliary verbs for fronted negative or restrictive adverbs (like 'Hardly had I...'), Swedish simply flips the subject and the main verb.