Is there a word or way to know if an interval is going from a low note to a high note? "Minor second" doesn't tell me if it's going from a low note to a high note for example.
Or maybe there's a way to designate direction with intervals?
It's very simple; we say something like "an ascending major second," or "a minor third below."
In analysis of post-tonal music, we have the notion of "directed" or "ordered" intervals, where we can say something moves +7 semitones followed by -9 semitones. This is in contrast to "unordered" intervals that do not specify direction.
The directed intervals of "Mary Had a Little Lamb" would thus be
-2 -2 +2 +2 0 0
whereas the unordered intervals, which are very similar to your example, would be
2 2 2 2 0 0
Yes. You just use the words up or down; e.g. major third up, minor third down. In writing sometimes an arrow is used.
You can speak of 'a minor 3rd up' or 'a minor 3rd down'. Or 'above' and 'below'. Or, if you prefer long words 'ascending' and 'descending' (though those should be reserved for serial intervals not simultaneous ones. The melody C, E is an ascending major 3rd. The chord C, E is a major 3rd above E.)
But if we're using interval names, they are always worked out from the bottom note up. Major 3rd up from C is E, because it's three letter names - C, D, E - and E fits into the C major scale.
We teach interval naming by 'is it in the scale? Then it's Major (perfect). Is it one smaller? Then it's minor (diminished). (Not a full explanation of course.)
However, confusing though it may be, C down to A is a minor 3rd. Yes, A fits into the C major scale. But when naming the interval C (down to) A, the reference point is the A major scale.