The fruit (the white kind) looked good for the first couple weeks, then one day the fruit split open from top to bottom and dried out. Was this from over watering?
It was probably caused by the same thing that causes this in tomatoes. It was under watered, then over watered, which causes the fruit to fill up and burst like a water balloon. This article explains how it happens in tomatoes and since eggplants and tomatoes are very closely related, it is the same thing: https://www.gardeningknowhow.com/edible/vegetables/tomato/tomato-cracking.htm
How deep do I plant, what type of soil, and how far apart do I set my eggplants? Also, how much sun do they need?
This article will help you:
https://www.gardeningknowhow.com/edible/vegetables/eggplant/growing-eggplant.htm
It's my first time starting eggplant seeds indoors. The first true leaves, though large, are starting to turn yellow. The 2nd to 4th sets of leaves are still dark green. The plants are about 4-5 inches and not leggy at all. Do I need to fertilize these young plants at all before it's time to harden them off?
The yellowing sounds like they need nitrogen, so yes they are tall enough to feritlize. Do the new green leaves turn yellow as well as they grow? If so then yes they need nitrogen. Still it does not hurt to fertilize them, all plants need to be fertilized or you can also mix some manure in with the soil. Do you plan to keep them in a pot as well?
My eggplant gets flowers, but then they fall off.
This article should help: https://www.gardeningknowhow.com/edible/vegetables/eggplant/what-to-do-for-eggplant-blossoms-drying-out-and-falling-off.htm
My eggplant seedlings have brown spots on the leaves. The spots eventually creat holes in the leaves giving them a lacy look. Then the leaves shrivel and die. What could be causing this and how do I treat it?
This is a fungus. There are several that it could be but they are all treated the same way. Use a fungicide on the seedlings. It will kill the fungus and the plants should recover.
My eggplant plant is huge and has gotten flowers for a while. What do I need to do to get the fruit?
It sounds like you may have a pollination problem. The flowers will not grow into fruit unless hey are pollinated. These articles will help:
https://www.gardeningknowhow.com/garden-how-to/beneficial/insect-pollination-process.htm
https://www.gardeningknowhow.com/garden-how-to/beneficial/creating-a-pollinator-garden.htm
Tomatoes and eggplant are pollinated in teh same way, so this article will help too:
https://www.gardeningknowhow.com/edible/vegetables/tomato/pollinate-tomatoes.htm
It is almost winter in Melbourne now and my eggplants have finished with the fruit. Do I pull out the finished plant or move them under my pergola, which is warmer, and look after them for next summer? Currently it is in boxes and in the open.
These warm-season crops will not tolerate any cold. After they have put out fruit and it is harvested, the plants will naturally die out to be replanted in the following growing season.