3. Are faith in God and scientific methodology mutually exclusive?
Yes.
The PURPOSE of scientific methodology is to remove "faith" from the results.
"Big rocks fall faster than small rocks, because they're heavier."
Have a little faith and common sense, this MUST be true.
Galileo Galilei invented the scientific method of experiment, by rolling different sized balls down an inclined beam, to disprove this and assert that all objects fall at the same acceleration, regardless of mass.
Feathers fall slower than hammers because of air resistance, as was shown by the astronauts dropping hammer and eagle feather on the moon.