Tomstar, greeting!
Welcome to the FES! I am personally neither a rounder nor a flatter. I am examining both from the perspective of one who wants to see logically which theory better holds. So far, from a Biblical and zetetic perspective, Flat Earth Theory (FET) wins. But from a scientific and mathematical perspective, so far, Round Earth Theory (RET) wins. I am leaning toward a Flat Earth understanding of the world, because I am inclined to trust my own eyes and the Scripture more than I am mathematical equations, most of which I don't understand anyway. However, I cannot just ignore Facts. And all the scientific evidence seems to point to a Round Earth. So I am in debate.
However, to answer your questions, What is on the other side of the Earth depends on which Flatter you talk to. Unfortunately, one of the weaknesses of FET is that there is no consensus on ANY topic except that the Earth is flat. Everything else is up for grabs. I will give you the answers I personally know, and am inclined to believe. In order to answer your questions, I am going to have to do more than that. I am going to have to give you the whole cosmology of my FET. So do be patient with me.
The Earth is indeed flat. Above the Earth is the Sky, or Firmament. The stars, the planets, the Sun, and the Moon are there. Above the Sky are the waters over the Firmament, which fall through the flood gates to Earth in the form of rain, snow, hail, etc. Above that is highest Heaven, where God dwells.
The Sky and the Earth meet where the mountains serve as the pillars of the Sky at the edge of the Earth. The pillars of the Earth are beneath the Earth and support it. This is where your question comes in. Beneath the Earth are the waters under the Earth. The seas and oceans surround the Earth, as the fountain of the deep (which is connected to the waters under the Earth). Beneath that is Sheol, the abode of the dead. Surrounding that are the waters of the nether world.
This is the ancient Hebrew understanding of the world from the Bible. As a traditional Jew, I am inclined toward accepting this cosmology of things. Naturally, the Earth is the center of the universe.
Although I am not certain of this view of things (given that math and science support a Round Earth), this is what I am working with right now. As to how thick the Earth is, that is a very good question. I don't know. And what happens, how does the Sky get held up by the pillars of the sky, the mountains? Well, basically, the sky comes down to meet the Earth. Exactly how it does that, I don't know. But there you are.
I hope this all helps. If I can be of further assistance, do let me know.