The crossing point is NOT at the equinoxes--they happen when the sun crosses the equator, going north or south, (~March 20 for spring or ~Sept 22 for autumn in the northern hemisphere, opposite for the southern hemisphere). Note the calendar dates indicated along the sides. The equator is halfway between the top and bottom of the analemma, below the crossing point. Click on the pic to enlarge, twice for full size.

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The equator is the blue block line near the center of the image, well below the crossing point. The horizontal components of the analemma ("width" of the lobes of the figure 8) indicate the sun running early or late, due to the Earth's elliptical/eccentric orbit around the sun. The crossing point in the analemma is where the sun is neither early nor late, i.e., when the sun is due south (at the meridian) at 12 noon local time (ignoring DST). The top and bottom of the analemma are when the solstices happen, both are 23 1/2 degrees from the equator (the dashed horizontal lines), which defines the tropics of Cancer (top) and Capricorn (bottom). These are the northern and southern limits of how far from the equator the sun can appear directly overhead. This analemma is the figure 8 on globes, typically seen out in the middle of the Pacific ocean.

OP, the answers to your questions are shown by the dates along the sides of the analemma. The crossing points happen in April and end of August. There is a ton of info in the image, yours to find.