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If the Earth's rotational axis was perpendicular to the orbital plane, but the orbit also highly eccentric (sun at center), how would that affect the day/night cycles?
What I think is that during the time the Earth's closer to the sun, because the gravitational pull is stronger, it would make the Earth spin faster than thereby creating much shorter days than if the Earth was near the focii of the orbit (on which the days/nights would be longer). Is this line of thinking correct?
Your thinking is incorrect. High gravity, or, more specifically, strong tidal forces lead to tidal locking not faster spinning. Moons are a good example of this. The Earth's gravity keeps the same side of the Mon facing the Earth at all times. The Moon still rotates but it's rotation is in sync with it's orbit around the Earth, 1:1, so the same size of the Moon (the heavier/denser side) always points towards Earth. Many moons are tidally locked to their planets as a result of the strong tidal force. Pluto and Charon are tidally locked to each other.
That said, tidal locking takes a long time. Just passing close to a massive object wouldn't stop a planet from rotating. It takes many orbits to create the tidal locking, usually millions or billions. Because planets tend to be pretty uniform in density, gravity doesn't affect their rotation very much.
A highly eccentric orbit, the sun isn't close to the center, either. A low eccentricity orbit, the sun is close to the center.
A highly eccentric orbit would affect seasons. With no Axial tilt, the seasons would be entirely governed by the distance to the sun, not the position of the sun in the sky which governs seasons on Earth as it is. To clarify, it's the Axial tilt that causes the seasons, not the distance to the sun. But as you specified with zero axial tilt, then the distance would govern the seasons, but at Earth's current eccentricity, that effect would be small. The greater the eccentricity the greater the seasonal variation in this hypothetical and the seasons would be planet wide, not alternating between the two hemispheres.
Summer would also be shorter than winter due to the Earth moving faster around the sun at the closest point,
The days themselves wouldn't be affected much. The length of a typical day (Synodic) would vary by a minute or two based on the Earth's orbital speed. It wouldn't be very noticeable but with a perfect 0 degree axial tilt, the 12 hours of sunshine would vary slightly based on orbital speed and position in the eccentric orbit. . . and the size of the sun in the sky and atmospheric refraction. . . . but only a couple minutes here or there with the Earth's current rotation speed.
