In our Beit K’nesset (largely kipa sruga), there are a few chabadniks who’s children spit on the floor during the Aleinu prayer. I find the practice disgusting, not because of a possible insult to the "goyim’s god" in the Aleinu, but because spitting, on its own is a disgusting practice and doesn’t belong in a beit k’nesset. What is your view on the practice? Is there anything wrong with requesting these spitters not to sit next to me in the beit K’nesset?

Spitting on the Beit Knaset floor during Aleinu is a common Chabad practice; they do not find it offensive or disgusting in any way. Therefore if someone who is personally disgusted enters their Shule, it is his own problem he will have to deal with privately. If a Shule is not a Chabad Shule and congregants of that Shule find it offensive they have the right to ask the spitters to stop. A possible simple solution is asking them to spit into a handkerchief or a tissue.