~ずに(は) - "without," the same as ないで: 何も言わずに・言わないで、部屋を出た。~ず comes from Classical Japanese, where it was a negative form. It's used in more formal speech and writing. ~ないで can be substituted in spoken language.
いられない - the potential form of いる, so "cannot be/exist." All together, "cannot be without ~," or as you put it "cannot help but be~"
This patter is usually used when someone does something in spite of themselves/their determination to suppress their emotions or actions.
ざるをえない means "no other choice but to~." This choice is usually against the speaker's because of pressure or the urgency of a situation.
