Hello and welcome.
I'm a round Earthling, but from the wiki and many flat Earthers this would be their answer:
• If the Earth was indeed flat, how could lunar eclipses be observed?;
There is many model flat Earth model for the Moon, the wiki of FES states there's a shadow object blocking the Sun from the Moon creating an eclipse.
• Where exactly is the South Pole on a flat Earth?;
Most flat Earthers doesn't believe the South pole exist, some uses a different flat Earth map where it do exist, like the bipolar map.
• How could the midnight sun (24 hour period of sunlight) occur on a flat Earth?;
The standard flat Earth map where the North pole in the center allow this to happen in the North pole, they would say midnight Sun
near the South pole is a lie.
• Some of you argue that gravity doesn't exist and the Earth is accelerating upwards. Wouldn't the Earth have to accelerate infinitely in order to prevent the resultant force on an object from reaching zero?;
Most flat Earthers who accept UA also accept Special relativity, so we have 0 relative to speed on the ground. This is the least accepted law of the flat Earth.
• How would a flat Earth orbit the Sun?;
The Sun circles above the flat Earth, it orbits closer to the South pole at winter, Equator at equinox and the North pole at summer.
• Finally, why would other objects in the solar system be spherical apart from the Earth?
Because the Earth is supposed to be special, and what better way being special than being uniquely flat?