This impossible to answer without describing the configuration the two suns, the earth, and any other planets are in. The answers could range from utter destruction to paradise.
I assume two suns the same as our current one. Well, that would mean two shadows for every one object here on Earth. Everything would have two of them. Earth would also receive double the radiation, so it would warm up. A lot. It may cause a runaway greenhouse effect, similar to what Venus went through, but this would take a very long time. Let’s move over some points that might be possible if the Earth had two suns.
1. We would have blue skin because it reflects the most energetic and harmful sunlight
If the earth had two suns a lot of scenarios would be possible. It would all depend on the orbits of the suns and their position relative to each other and earth. Their relative masses would also matter.
Assuming that therefore we would have more energetic and harmful sunlight is not necessarily true then.
But if we do, that doesn’t even mean our skins would be blue. If blue skin was such an evolutionary advantage, it may have developed by now in our present solar system.
2. More happiness because we’d have an increase in vitamin D
This is not my terrain but the fact that a vitamin D deficiency makes you feel bad doesn’t necessarily mean that an overdose would make you feel good.
3. A year would be 493 days and a month 50 days.
Huh? I wonder where those numbers come from. This would all depend on the orbits of the suns and the earth and moon. It could be that, possibly, but it could be many many other numbers.
4. Many seasons, short nights, great temperature range
Many seasons would depend on the orbits and the angle of the earth’s axis to the orbital plane. Also, would the earth be tidally locked to one of the suns?
Short nights again depend on the orbits and position of the suns and on earth’s rotational period. If the earth would be tidally locked a day-night cycle may take a year, half a year, two years per three days… anything.
Great temperature range depends again. If you would work on the assumption that the one sun can rise while the other sets, there would not be more range in temperature. Predicting the climate effects is very hard unless you just assume that the two suns are doubling (or just increasing would be enough) the solar energy showered on the earth, in which case we’d be fired.
5. Two shadows
Probably, but it depends on the orbits.
6. Plants more flourishing
If you conclude that there will be more sunlight then yes, probably, but again it would depend on the orbits and the luminosity of the suns.
7. Triple eclipses
Depends on the orbits and their inclinations
8. Moon would pass between the two suns
I find it hard to see how that would happen since the moon’s orbit is less than 400,000 km from the earth. If any of the suns would be closer to us than that we’re not gonna survive that.
But if you mean that we could see from earth a moon in a location between the two suns, then yes, that would be possible. Again, it would depend on the orbits.
Here are two possibilities to consider:
The suns orbit each other closely and earth (b) orbits around it:
Earth orbits the first sun like now and there’s a secondary sun in a wide orbit around that: