Does the emission of semen necessitate ghusl—even if without pleasure—if it occurs continuously, and likewise, if one feels its emission after urinating or when desiring to defecate?

What you have described is not maniy (sperm/seminal fluid) which necessitates ghusl (ritual bath). It could be wady (a thick, white discharge that comes after urination) or normal vaginal discharge, neither of which necessitates ghusl. Ghusl is only required when the emission of maniy is confirmed with certainty. If there is doubt, it takes the ruling you wish. One should be cautious of waswas (obsessive whispering/doubts) and not succumb to them. What is emitted of maniy due to cold or illness, without a gush or lust, does not necessitate ghusl according to the majority of scholars.