Learn Swedish: Barnet kan inte vara tyst i en minut. - The child cannot be quiet for a minute.

You are right that negative sentences usually take på to mean "for" (like Jag har inte ätit på en timme - I haven't eaten for an hour). However, på is used when an action hasn't happened at all during that time. In this sentence, we are talking about a continuous block of time ("for the duration of a minute") that the child is incapable of filling with quietness. For how long an action lasts, we use i.

Could i en minut also mean "in a minute" (like "I will be quiet in a minute")?

No, to say "in a minute" when talking about the future, Swedish uses the preposition om (om en minut). When you use i with time, it almost always means "for" a specific duration.

Since barnet is an ett word, why isn't tyst spelled with an extra 't' at the end?

Adjectives usually do get a -t at the end when describing an ett word. However, since tyst (quiet) already ends in the consonant cluster -st, adding another -t would make it too hard to pronounce. So, it just stays tyst.

Why does inte go between kan and vara?