"Kepler noticed that the farther Mars was from the sun, the more slowly the planet moved, and the closer it was to the sun, the faster Mars moved along its orbit.
This also happens with the use of epicycles and Ptolemy’s equant. Kepler was well aware of this. The problem was that this model only came close to accuracy, which is why new planetary tables had to be produced.
Tycho was convinced he could falsify the Ptolemy model by using the relative distances to Mars. Many of Tycho’s measurements were accurate to 1/2 arc min — 2x better than the eye can see.
Kepler knew he had the basis to produce an improved model — Tycho’s data for Mars.
This high confidence in Tycho carried Kepler for years in doing his heavy math calculations involving hundreds of pages of calculations.
1) How did he measure the distance of Mars from the Sun?
He didn’t, AFAIK. The transit of Venus gave us that, though there were many reasonable other approaches.
2) Since Brahe and Longomontanus were geocentrists, presumably they did not buy into Kepler's observation, correct?
I’m guessing Tycho, at least, had died before Kepler conducted measurements. I also think that Kepler’s observations were useful only after sneaking a telescope from Galileo, via a friend. [ Galileo, apparently, was concerned Kepler’s known bad eyesight would bring heavy doubt on his new invention.]
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