First off, the ring-shadow would be seasonal. On the northern Summer Solstice, the ring-shadow would fall on the southern hemisphere to the fullest extent. Near the Equinoxes the rings would be lit mostly edge-on. (For the math types, think about the sun's angle at noon wrt the Equator over the year as a sine wave. Then change in position is the derivative of sin(x) which is just cos(x).) In short, the sun's angle changes the slowest near the Solistices and fastest nearest the Equinoxes. That means locations near the Equator should be minimally affected by a few yearly ring-shaded days while low-mid latitude locations would have a significant number of mostly ring-shaded days each year.
Given the ring width in the video, it looks like most ring-shaded locations would have at least part of the day in direct sunlight. Also, because the shaded days occur around the winter solstice, plants already in hibernation shouldn't have a problem with the lack of direct sunlight. So, temperature would probably be the main concern for life. The ring-shadow should make winter colder and start cold weather earlier in the year. I would suspect weather patterns to be affected, but I couldn't even begin to predict those sorts of changes. However, at a global scale we're talking about blocking some reasonable percentage of incoming sunlight (10% per day near the solstice maybe?) so global temperature would almost have to be lower with an impact that large.
One other note. The ring system wouldn't be stable with our moon - tidal forces would distort the ring shape until ring particles hit the atmosphere and fall out of orbit. So I'm afraid we'd have to choose one or the other. As pretty as a ring system would be, I think it would be bad for our satellites and make it that much harder to send people into space. So, my personal vote is that we keep the moon.