But it doesn't mean "claim it as your own", as long as it's not significantly altered.
No, you cannot relicense. Only the original authors can do that. You can add on top of the existing code stuff of your own under a different license, so that the combination of the two is under both licenses. Practically, it amounts to almost the same, but for example, you cannot remove conditions such as attribution, which only the original authors can do via a license that does not require attribution.
It depends what was meant by derivative and how this impacts the original code (encapsulated, modified, etc...). I was thinking more of when you use an MIT-licensed lib or 3rd party tool, or when you would reuse a code-snippet licensed as MIT and modify it. License and attributions remain, but only for that part of the code.