I know heavy armor ignores Dexterity modifiers for AC calculations, but what about medium armor for a character with a -3 to their dexterity? Does Medium armor bottom out at -2, or will -3 still apply?
No, you could still apply a -3 modifier
This is an understandable question, because the table of armors in the PHB (p. 145) states that you add your Dexerity Modifier "max 2" to your medium armor AC. This is a tad ambiguous, and could be read several ways. However, the rules on medium armor spell out the rules on modifiers a bit more explicitly than the table of armor does. In PHB p. 144, it states (bold added):
If you wear medium armor, you add your Dexterity modifier, to a maximum of +2, to the base number from your armor type to determine your Armor Class.
Since this clause specifies a "+" 2 as the maximum, it clarifies that what is meant by "max 2" on the armor table is that a positive 2 is the highest possible modifier. Thus, the "max 2" on the armor table doesn't apply to modifiers that are below 0 (and thus below the "max" of +2).
Contrast this with the rules for Aura of Protection, which states (PHB, p. 85, bold added):
the creature gains a bonus to the saving throw equal to your Charisma modifier (with a minimum bonus of +1). You must be conscious to grant this bonus.
In the absence of a stated minimum modifier, the -3 "bonus" from a Dexterity of 3 or 4 could be applied to your AC if you were wearing medium armor.