Specifically, do the folks who figure out trajectories and forces, rocket timings, etc. to get an object to orbit the moon also take into account the gravity of the sun as they plan? What about the other planets? Is their pull significant?
To the Moon? No. It’s a pretty big throw to get something there, but our sun really doesn’t figure in there.
Keep in mind that the bulk of the calculations done in the 1960’s were done using sliderules and trig tables. Just figuring the acceleration of a rocket that is constantly losing mass (fuel) and leaving a square-law gravity well is fairly complex when you can’t approximate it with a gazillion piecewise linear segments on a digital computer. Crude as they were, the Apollo era computers were as critical to the mission success as the rocket engines.
They didn’t know it at the time of the first Moon missions, but predicting lunar orbits accurately is much more complex than you might think. Due to the presence of mascons (mass concentrations), the Moon has a noticeably “lumpy” gravitational field. This can perturb the orbits of satellites in strange ways, and cause them to crash into the lunar surface far earlier than would be predicted by a simple point-source model of the Moon’s gravity.