I'll try to fit my answers to your follow ons..
a') the six months apart example for parallax was for nearby stars in our galaxy (only) because they don't tend to move over 6 months. [it doesn't work for anything else astronomical like galaxies or far off stars within our galaxy]. For Venus, and Mercury, they move very fast relative to earth observations, so taking the measurements on Earth are time sensitive. But taking them at the same (approximate time, at different vertical (latitudes) gives us the geometry for our solar system that we would need.
b') the transit of a planet by the sun
HAS TO be approximately when the planet is at its nearest point to us in its orbit, but not necessarily when the Earth is at its nearest point in its orbit; it is at anytime in the Earth's orbit.. right. At any time in the Earth's orbit, Venus could eclipse/transit the sun, and that makes it approximately 0 degrees with our line of sight with the sun. It is the difference between the approximately and the absolute that makes this work. Both orbits are ellipses (not circlular orbits), so the distance will vary of course. In the case of Mercury, the "ellipse" precesses in space; the famous "Precession of the Perihelion of Mercury" [You know what precession is.. it is what happens to a top when it loses steam and wobbles before it falls] that was observed since Newton's time, but only explained by Einstein's General Relativity (see
https://aether.lbl.gov/www/classes/p...ionMercury.htm)... but I digress.
c') yes, I think you have it. Remember, since Galileo's time, the points of light in the sky became disks that had areas. People could look at tops and bottoms. Once they could resolve these things, then geometries and measurements could be made to Copernicus's decentralized solar system. Exciting stuff! Back then, they were really excited about different ways to measure the diameter of the earth (Kyrie missed that one in the history books). So it was natural to look out to the stars (the planets being the closest of those celestial bright things).
d') As mentioned in a', yes times did make a difference for planets that move fast. Perhaps not by the day, but by the hour if you wanted significant digits and you were not along the same meridian. It takes Venus 3 to 7 hours to move across the face of the sun (depending on idiosynchracies of the orbits). My sun dial doesn't show seconds.
Larry
DevilHorse