It's not controlling anything, just providing steering cues for the pilots along whatever route it's set up for. Setting it up for the flight's intended route is a checklist item, which appears not to have been performed on this flight. Also on the checklist is identifying the correct airspeed for current conditions, called "Vr," at which to "rotate" and bring the aircraft off the ground during the takeoff roll.

Actually doing that is a memory item, intended to be prompted by a "Vr" call from the pilot not flying. As far as I know it's not yet clear whether the PNF gave that call, but the PF certainly waited to rotate until well after reaching Vr. This is where the "sanity checking" appears to have failed, with the PF heeding the misconfigured flight director when it disagreed with other instruments and the situation visible out the cockpit windshield.

It's heading, altitude, and speed. For example, if you're currently accelerating to a target speed and climbing to a target altitude, it won't command you to pitch up so much that you stop accelerating.

The autopilot follows the flight director, and the flight director is dumb.

The autopilot wouldn't have been in use for the takeoff roll, though, would it? The question of automation smarts seems like a bugaboo in what I understand to be a highly manual flight regime.

Correct, autopilot isn't engaged until at least several hundred feet AGL; varying by airline. Some airlines don't engage autopilot until reaching transition altitude or even cruise altitude. Still, the flight director is giving you the same information, and most pilots will just follow that because it's the most accurate way of getting you to where you need to go... if you've set it up correctly.