Galileo's proposal involved circular heliocentric orbits. (Kepler would later improve Galileo's theory by switching to elliptical orbits instead of circular ones.) The orthodox geocentric or Ptolemaic theory of the day involved "epicycles" to explain retrograde motion, and actually did fit the data better than Galileo's theory.
As far as the Catholic church is concerned, what got Galileo in trouble was not heliocentrism per se, but Galileo's tendency to insult anyone who disagreed with him, and some remarks of his which seemed to say that, if his theory disagreed with the Bible, then the Bible was wrong. Heliocentrism was never formally condemned as heresy. (Actually, Galileo was never formally condemned as a heretic, either.)
From the POV of modern science, it's really making a bit of a mountain out of a molehill. The earth doesn't rotate around the sun, and the sun doesn't rotate around the earth. They both rotate around their common center of gravity. "Heliocentric" and "geocentric" refer to where the origins of coordinate systems are placed, but those are just arbitrary conventions selected to simplify the math. For example, when NASA computes the trajectories for earth satellites, they use geocentric coordinates. When they compute the trajectory for a Mars mission, they switch to heliocentric ones.
Absolutely. - In fact Ptolemy's model is still in use today in stellar navigation, and surveying. The U.S. military still teaches the method in the military academies, and also OCS. When GPS is down, it is the best way available.