1. Why does the cold air comes down to the valley here but nowhere else?
2. Is the reason why the air at the top cooler than the air at the bottom of the valley because of the lapse rate?
3. Why does the top of the valley (mountain) get warmer than the bottom of the valley? Slope, Aspect, etc matter too?
If you’ve ever seen liquids of different densities form distinct layers that’s basically what’s happening with the air. Cooler air is denser than warm air so it wants to sink while warm air wants to float to the top. They’re made out of the same stuff but the temperature makes them act different from each other. On a flat area the ground is warmed by the sun and that heats the air close to the ground. This is how it works in most areas, so during the day the air on the ground is the warmest part and it gets cooler the higher you go. In a valley with steep sides part of the ground will get sun on it so there’s some air warming but there’s still a lot of cooler air from the night before that wants to sink to the bottom. The higher-up parts of the valley get more light so the air is warmer the higher you go.
Air flows just like water flows. Usually need some fog or cloud to see it while it happens, but it is happening whether or not you can see it. So, denser air sinks down and less dense air rises.
Cold air is a bit more dense than warmer air, so it tends to sink, to flow down and under nearby air that is slightly less dense because it is a bit warmer.
Ideally, air density is solely a function of pressure, so the higher up you go, the less dense air becomes, and it also becomes cooler because it costs heat to expand (uses energy to decrease density; also it is sending heat up and getting heat from below so a steady-state condition arises even if no air moves; warmth migrates from below and goes to the cold of outer space). The real world is a lot more complicated because air refuses to sit still.
Air is really fluid, so it moves (convects) heat around quite a bit more than it transfers by conduction or radiation. This is why there is so much wind, because the air heats differently from location to location and moves in big masses. Also, as a gas, it changes density fairly easily with modest changes in pressure (unlike water).
Inversions generally require a physical trap of some sort to exist, a basin or a depression that can fill up with denser air, much like a pond on land fills with water. Cold air descends into the valley and fills it, displacing the warmer air that was there, which rises up, resulting in a basin of unusually cold air with a cover of warm air. You can get inversions even without a physical trap but it requires odd air temperature patterns, like a warm front where hotter air rises above a mass of colder air, trapping cold air down near the ground.
A funny (odd) thing about air is that it warms up a bit when it compresses. This can create abnormal warm winds when dense cold air descends a good distance, so increases in pressure quite a bit. That pressurization is work being done on the air and it makes heat, just like pumping up a tire causes the air to warm up. These oddly-warm winds are called Chinook winds out in the plains near the Rockies. If air falls from several thousand feet altitude while crossing a high mountain range, it will warm up as it descends rapidly on the near or down-wind side of the air dam that is the mountain. The air higher up will be cooler than down below. At least up to where the inflowing dense air is still being capped by a layer of warmer air. The temperature will rise in the zone of contact between the pool of cool air in the valley and the warmer air that is above the mountains, but it behaves as would be expected (cooling slowly from ground upward) until it hits that contact.