Suppose The Earth Got Saturn's Rings...

That was my exact thought too - and from there probably some other conclusions too.

Not sure thats true.

http://www.lhup.edu/~dsimanek/flat/flateart.htm

As per this website (and what I can remember) the first evidence that the earth isn't flat came from the way ships disappeared as they got far away.

I guess some ancient cultures may have seen the shadow of the top of the earth on the ring looking round, but they might just as easily have come to the conclusion that as you go up north (or down south) you get to a big mountain (Mount Olympus maybe)?

As far as I know our comprehension of brightness is logarithmic. Something with a ten-fold increase in brightness over the moon, would in reality look barely any brighter.

I wouldn't doubt that if the earth had a series of rings that we would still be able to see the stars. However, this would obviously depend on the composition of the rings and their albedo. Also it would vary by latitude, at the equator the rings would be barely visible.

Latitude would have likely been easily discernible far sooner than the usage of astronomy in navigation, it wouldn't be hard to measure the width of the rings to measure your latitude. Even a crude measurement would have been immensely useful to early ships crews. Longitude would likely have been more problematic, however knowing the moons phases and the time of day (which would be discernible by the position of Earth's shadow on the rings) would likely be enough to determine longitude.

I doubt human development would have been greatly affected by having planetary rings, it may even have helped the technological revolution by enabling long-distance sailing earlier.

I think your right. The tides are caused by the fact that the earth spins as as result, the moon moves relative to each spot on earth. As long as the distribution of mass never changes (e.g. if its always uniformly distributed), any gravitational effect they have on each part of the earth would be constant.

However, I have no idea if the rings of Saturn are uniformly distributed or what.

Cold deserts?

Here are a few basic thoughts given some physics and geometry arguments.

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.

What I want to know is what things would be like if we were a tidally locked moon of a gas giant, which, due to the ambient heat of the giant and proximity to the sun was still livable.

There was a scifi book about this - cannot remember the author - but the satellite was close enough to the gas giant that it was extremely elongated.

I'm more curious about the "what if" scenario - what if it were more spherical than spheroid (like earth); what if it was livable (at least on one hemisphere): what would the seasons be like? what would the weather be like? Given that it's tidally locked, the day would be determined by its orbital period, not its rotational period; how would this affect the length of the day, especially if it has an orbit which is not parallel to the gas giant's orbital plane? Eclipses would be much more prevalent, I'm sure.

How would people on such a planet determine their calendar? Phases of the planet? Hmm.. It would be interesting to seriously calculate all of this out. Unfortunately, my experience in astronomy approaches nil asymptotically :)