Was standard Newtonian mechanics sufficient or were relativistic effects included?
Relativistic effects didn't have to be modeled; other sources of error
would have swamped the effects of relativity, and midcourse corrections were made.
Were the Earth, Moon, and spacecraft modelled as point masses or more complicated bodies?
The moon's gravity was modeled with the L-1 potential model consisting of "5 coefficients out to a maximum degree of 3". I don't know a lot about geopotential modeling; it's discussed a little bit in this Q/A and the "Tindallgrams" linked therefrom. Apparently the model was updated between Apollo 11 and 12 with better data obtained from Lunar Orbiter data.
A gravity field is modeled (and can be visualized as) deviations from the gravity of a perfect sphere. The visualization of the L-1 Lunar potential looks like this (with red indicating increased gravity and blue decreased):
And the Apollo 12 gravity model looks like this:
And, for reference our current lunar gravity model, based on GRAIL data, looks like this:
The spacecraft's attitude control system required good estimates of its mass, center of mass location, and moment of inertia in order to maneuver efficiently. Without that level of detail in the modeling, the attitude control system would likely have been less responsive or more wasteful of propellant or both.
I'm not sure about their Earth gravitational model; they may not have needed much detail since they would only be in parking orbit for a few hours on the way out and re-entering aerodynamically on the way back.
I don't believe solar wind was factored into navigation, again because any effect it would have was easily corrected for.
Approaching the question a different way: to what extent was the control of the spacecraft pre-calculated, vs. calculated in real time by spacecraft computers, vs. done manually by the astronauts?
Mostly navigation was done by computers on Earth. The position of the spacecraft was accurately tracked throughout the mission, and correction maneuvers would be calculated on ground-side computers and called up to the crew to execute as needed; the guidance computer would execute the maneuver with a crewman ready to hit the shutdown button if necessary.
A few parts of the flight were flown manually. The terminal phase of the moon landing was one of them; the capability existed to land semi-automatically using a radar altimeter, with the commander able to adjust the targeted landing spot, but in every landing the computer was switched into a more manual mode at around 500 feet altitude and landed with the commander's hands on the controls while the LM pilot managed the computer and called out altitude and speed figures to the commander. Lovell intended to try to use the automatic mode on Apollo 13, but didn't get the chance to attempt the landing.
The transposition-docking-extraction maneuver to pull the LM away from the booster was flown manually by the command module pilot (CMP).
After the LM separated in lunar orbit, the commander would manually turn the LM to allow the CMP to visually inspect it; on the LM's return from the moon's surface, the commander would manually fly the very last part of the approach and docking, with the CMP ready to take over the active role if a problem developed on the LM.
