Dtyjlui

Say if I play G to C descending, my intuition tells me to start from the note that came first. But I've been told that's not correct. You start from the bass note and count up no matter what.

You can describe intervals as ascending (lower note to higher note) or descending (higher note to lower note), although almost all music theory concepts strictly talk about ascending like your teacher says.

The reason for this is simple and it's to do with the physics behind music. Notes generate what are known as harmonics. So not only do you get the fundamental pitch, but you also get multiples of it depending on the exact tone. This is also why inversions, even though they contain the same set of notes, can have different functions and roles in harmony due to the interaction between them. It's also why we make distinctions between inverse intervals. For example, the perfect 5th interval's inversion, a perfect 4th, is dissonant in some contexts even though the perfect 5th has been consistently considered consonant.

For more reading about the importance of bass in harmony see this question.

It could have been counted either way - resulting in two different answers to the same question - what's the interval between G and C?

It's easier to count upwards, and most of what we do goes left to right, so this has become convention. G>C we count as P4, whereas C>G is P5. G>C descending puts C underneath, therefore it's P5. There's a 'rule of nine' for inverts, so P5 inverted becomes P4. Majors become minors when inverted, and diminished become augmented (and vice versa). Agreement on the method used to identify intervals saves a lot of arguments and confusion!

Although, as @Dom says, the convention is to describe the ascending interval, it doesn't really matter in terms of distance if you describe it ascending or descending. It becomes clear if you use a specific pitch label like Helmholtz or scientific notation. G4 to C4 is the same as C4 to G4. Either way it's still a perfect fifth. But the convention is to refer to a G4 above a C4, or a perfect fifth above C4.

Just to add a bit to the detail about bass and harmony: when describing chord root movements (not actual bass movement) the convention is describe descending intervals from a step to a fifth and then ascending intervals for anything larger than a descending fifth. For example, root progression by descending fifth or descending third, but rather than root progression by descending minor seventh it would be root progression by ascending major second. Not root progression by descending sixth, but root progression by ascending third.

For harmonic intervals, saying "a perfect fourth" covers both directions: a perfect fourth above C is F, and a perfect fourth down from F is C.

For melodic intervals, you would start from the first note, not the bottom: I go up a perfect fourth from C to get to F, and I go down a perfect fourth from F to get to C. It's about the direction you're going, not which one is higher or lower.