Absolutely agree with all of this. One thing I'd add: treat every task, even the menial ones, as a case study in how the agency actually runs. The creative intern who understands the business side of things is the one who gets remembered when full-time roles open up.
The weekly log is non-negotiable, by the way. I've seen too many talented interns walk out with nothing but a handshake because they couldn't articulate their impact. Documenting wins in real time makes your case for you when the conversation turns to conversion.
And on the mentor point - don't just pick the most senior person in the room. Pick someone who's three to five years ahead of you. They remember the grind, they'll be honest, and they're usually more accessible.