How are summer and winter solstice, autumnal and vernal equinox related to the ecliptic &celestial equator?

Answer 1

Equinox
"equal night" (meaning night has same length as day) This occurs (mathematically) when the sun's declination is exactly 0: i.e., when the sun is on the celestial equator.

The points where this occurs are the intersection of the celestial equator (declination 0) and the ecliptic (apparent path of the sun around the celestial sphere).

Length of Day is calculated with the equation:
Cos (P) = - Tan(Lat)*Tan(Dec)
where P is the polar angle at sunrise and sunset (therefore 2P is a whole day).
When either Lat or Dec = 0, then Cos(P) = 0
therefore P = 90 degrees
and 2P = 180.

The sun's polar angle P changes by 15 degrees per hour, therefore it takes 12 hours for the sun to go 180 degrees (leaving 12 hours for the other 180 degrees below the horizon).

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Solstice ("Sun stands still") is the point where the ecliptic is furthest away from the celestial equator. When the sun reaches that point, its declination ceases to increase and begins to to decrease. For a mathematical moment, the rate of change of the declination is zero (the sun's declination stands still).

If the Earth's orbit were an exact circle AND if Earth's rotation's tilt was perfectly fixed (there are minor wobbles), then a solstice would always be 90 degrees away from an equinox.

Everything is calculated form the "Vernal" equinox (the equinox that marks Spring in the northern hemisphere -- in March). Because of the precession of Earth's rotation tilt (a period of 25,800 years), the position of the Vernal equinox (intersection of ecliptic and equator) changes by 50.29" every year.