I have to say, I am surprised so many people have problems with eggplant. It is very reliable for me. It is not a fast grower and it does like heat. Like peppers, it does not germinate until it is warm, like 70 degrees reliably. If the plant is already growing it will tolerate cooler temperatures, but seeds won't germinate.
Because they are slow from seed, I plant them in starter pots using my standard mix 50/50 peatlite with some osmocote slow release fertilizer. I know you are probably using an organic mix. I don't have luck with those. They hold too much water and will rot my seeds. What every you use it needs to be fast draining, moist but not soggy and dry out in a day.
Out of habit, I plant nearly all of my eggplant and pepper seeds uncovered or lightly covered in the media, but I do have to cover the tray to keep the birds from eating the seeds.
I have some black beauty and long purple (pintung and Waimanalo long), but my favorite are long green ones. I prefer the thai long green, but I have to find the right variety. I have Roleks now, the skin is still tougher than I would like. The advantage of the green eggplant is that it can stay longer on the plant and still be soft enough to eat, pretty much when the purple eggplants lose their shine they are too hard and seedy to eat. I also have thai tiger (small round), it has more bitterness and is not a long keeper and it is literally two bites. Sometimes green eggplant is a hard sell because people don't like the color.
I transplant them when they get fairly large true leaves which can take a month, then I transplant them to 4 inch pots where they can stay for another month. Finally they would go into a 20 inch pot or the garden. In all it takes close to 3 months for the eggplant from seed to first fruit. However, once they start to fruit, I get 8-10 eggplant every 10 days or so until the days get shorter and the eggplant stops blooming. One plant can live for years, I know someone who has one that is still producing at 8 years. I generally keep mine in a pot so I replace it when it becomes less productive in about 3-4 years. I only keep one or two plants or I won't be able to eat or give away enough of the eggplant. My mother has a wild green eggplant that is very productive but it must have been a cross between a domestic and a wild eggplant because hers is thorny and mine is not.
I would not be discouraged if your plant is only a couple of months old and it is small. That is how they are, once they get past a certain point they just take off. I put a relatively small plant in my 20 inch pot. I know even though there isn't much top growth that the roots are expanding. Eggplant, like tomatoes need a fairly large root system to support the plant. If my pot is too small the eggplant will stunt.
I could and do have wild eggplant come up in my acidic garden with a pH 6.4, but eggplant needs to go in as transplants when the pH is 7.4 or higher. They are happier in more acidic conditions. The plants are fuller and not so leggy. They like a well drained soil but also like lots of water. I feed my plants about once a month in pots, but I do not use organic food, I use citrus fertilizer. You could probably use tomato tone. Eggplants are a small shrub so they need room or they will get fungal diseases on their leaves. If that happens, it is easier to head the plant back and let new leaves grow.