"Where do you guys go to school?"
Good question. My experience is that Chinese sounds better in paired syllables/characters. So often the times when they drop a syllable is both when they don’t need it and when they can group the characters differently into pairs. ... So this sentence already has shang xue (attend school) as a kind of paired set and would sound weird with an extra syllable/character. However, I would say qu xuexiao (go to school) where the longer word for school is the pair and qu is separate. None of this is rules, just a sense of flow, not sounding clunky ... But i’m not an expert so keen to hear what native speakers would say.
Yes, what you said about 2 syllable words is true.
However the 学 here is not really a shortened form of 学校. Compare with 上班, where 班 does not mean Company either. It is not usual to say 上学校 for "go to school (to study)", but it is alright to use 上学校 for saying "go to the school" for other general purposes (still less usual than 去学校 though). We say 上 for some places much often, e.g. 上教堂，上廁所. There is no rule for the preference, which can only be learned.
Usually 学校 is shortened to 校 rather than 学. 军校/Military school, 校际比赛/Inter-school competition, 我校/Our school.