|MadSci Network: Astronomy|
I'll take your questions in order:
1) Weather patterns and structures in Earth's atmosphere are dominated by the Sun, and Solar heating. The airflow driven by solar heating is also heavily modified by surface features on Earth (mountains that block airflow, bodies of water that store heat, etc). The Moon's tidal force plays a much weaker role in modifying the structure of our atmosphere.
The Moon does, however, dominate tides on Earth. The Sun also causes tides on Earth, but they are smaller than those caused by the Moon. As a result, the effect of the Sun on tides is primarily seen as a modification of the extremity of the tides caused by the Moon. When the Sun, Moon and Earth are aligned (New Moon or Full Moon) the tidal forces of the Sun and Moon act together, and we get the highest high tides and the lowest low tides. These are called Spring tides (I don't know why). When there is a right angle between the Sun, Earth, and Moon (First and Third Quarter Moon, or what we'd also call a "half" Moon), the tidal force of the Sun partially cancels that of the Moon, and the tides are at their least extreme. Those are called Neap Tides.
In determining the atmosphere and tides on Earth, we can pretty much ignore anything other than the Sun and Moon.
2) A tsunami is caused by a very rapid disruption in the ocean floor, usually an earthquake. If an object the size of the Moon were to pass by the Earth, at the same distance as the Moon, it would certainly affect tides on Earth. It would probably cause tides comparable to those caused by the Moon, but they would follow an unusual pattern, determined by the motion of the intruding object.
You might get some stronger effects in places where the local geography was just right. For example, right now, the Bay of Fundy in Nova Scotia has just the right shape to have a resonance with the lunar tides. This means that the tides are "pushing" the water around at just the rate that it would move naturally in the bay, which makes the tides there higher than usual (you can do the same thing in the bathtub by moving back and forth at just the right rate). Tsunamis would not result, however, because the disturbance would occur over a time of days, not seconds.
On a side note; comets don't come as big as the Moon. A typical comet is less than 10 miles across. The gravity of an object like that is so weak that it could have no effect on tides. The only way that such objects can significantly affect Earth is to hit it (that would be a disaster).
3) I'm not sure what you're asking about moons and rings. We believe that Earth's moon formed during the early phases of Earth's formation, when it was hit off-center by a Mars-sized object. The collision knocked a stream of debris into orbit. Most fell back to Earth, but some rocky material stayed in orbit, and formed the Moon. No ring was involved.
The Jovian planets (Jupiter, Saturn, Uranus and Neptune) do all have rings. Their rings, however, are all inside of their Roche radii. The Roche radius of a planet is the critical radius, inside of which the tides caused by a planet are so strong that a moon could not hold itself together. As a result, Saturn's rings, for example, could never pull themselves together gravitationally to form a moon. Tiny moons can exist within the Roche radius of a planet, but they are held together more by chemical bonds than by gravity.
[Moderator's note: Stephen's answer is completely correct. I'd just add that the Sun also has a Roche Limit, as does any massive body. However, its Roche Limit would only be a few times its radius, so would only extend a few million km above the Sun's surface. By comparison, the closest planet to the Sun, Mercury, is about 60 million km from the Sun. ]
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