Dr. Seuss: to address both your questions: in answer to your first question, the rebbe zya had no successors as he had no kids. It is also said that the chassidim didn’t feel there really was someone who was fit enough to take over the legacy.
In response to your second question: the proper name of the chassidim is lubavitcher chassidim as when it left europe (russia really) that is the city it left from. As to why there is a title chasidei chabad is because it is a chassidus based on intellect rather than emotion (chasidei chagas). Chabad = chochma, bina daas (the first letters of each in hebrew). Chagas /chagat= chessed, gevura, tiferes.
Stuck: no. What I think the view resulted from is that he had no successors unlike the frierdikker rebbe zya (who some actually did think was going to be moshiach) who did.