In particular, does the Autopilot in a B737NG control the pitch only via trim adjustment, or first via the elevators and then trim?
This is from the B737NG airplane manual. Boeing doesn't offer the technical drawings in the pilot manuals, one would have to have access to the maintenance manuals. Boeing is very sticky about their material being published on the internet.
Autopilot Flight Director System (AFDS)
The AFDS is a dual system consisting of two individual flight control computers (FCCs) and a single mode control panel. The two FCCs are identified as A and B. For A/P operation, they send control commands to their respective pitch and roll hydraulic servos, which operate the flight controls through two separate hydraulic systems.
This is a brief explanation of an autopilot system such as on the B737 from this website.
Aircraft with hydraulically actuated flight control systems have autopilot servos that are electro-hydraulic. They are control valves that direct fluid pressure as needed to move the control surfaces via the control surface actuators. They are powered by signals from the autopilot computer. When the autopilot is not engaged, the servos allow hydraulic fluid to flow unrestricted in the flight control system for normal operation. The servo valves can incorporate feedback transducers to update the autopilot of progress during error correction.