Can you display in Connect how long you've trained in each heart rate zone per week, month, etc.?

Not that I know of. But in Velo Hero, you can view your current heart rate zones, the last seven days, the last 28 days, the week, the current month, the year, and all. In intervals.icu, you can view the current, month, 42 days, three months, six months, and the year.