The drain of my tub has a pop-up stopper, operated by a lever on the overflow plate.

1. My first question is, what is the appropriate way to clean this kind of drain, when it gets clogged with hair?

2. My second question is this: in order to clean it, I have been pulling the stopper and rocker arm out of the drain by loosening the overflow plate. Most recently, when I did so, I pulled the stopper and rocker arm all the way out, disconnecting it. This was helpful because I managed to remove hair that was much further back. However, now I don't know how to reconnect it. How do I reconnect the rocker arm with the vertical linkage?

It does NOT "reconnect". You lift the spring with the handle then the rocker arm goes into the drain and fits UNDER the spring.

I've tried feeding the rocker arm back in, but the stopper remains down when I operate the lever.

Completely remove the overflow plate with attached linkage. Most will have a spring on the end where most of the hair gets caught. With plate removed, reset stopper and arm, then reinstall plate/linkage.

I've just tried this and the stopper still stays down when I operate the lever.

Can you send a photo of the linkage?

All we can tell you is how it SHOULD work. If it does not, then you have some unusual problem that we cannot diagnose without being there.

Most tub stoppers of that style are a ground joint mating system. Did you add the gasket or was it always on the stopper?

The gasket was always there.

A solution would be to replace your current tub drain with a "Tip Toe" drain. You press it to close it and press it again to open it. Very simple and easy to clean hair out when necessary.