Just let it grow. It should does not need to be trellised. How are the bees in your area. Maybe the blooms are falling off from not being polinated. If that is the case, take a small paint brush and touch the pollen from the male flower, and them the female flower and polinate it yourself.

Eggplants are like tomatoes, self pollinating. Don't need bees, or human imitators. Like tomatoes they may drop blossoms when conditions are not right for procreation. Just relax and be patient, barring predators and poor environment, they will take care of themselves nicely. Easy plant to grow as long as it has good soil and is not crowded.

Yep, let it grow. I tend to stake mine or put them in a small tomato cage, but it's not really necessary. They are self-supporting, though I've had the Japanese type bend quite a bit when laden with fruit.

I will stake one now and then if it seems to have trouble standing up on its own but pruning isn't needed for any reason.

You might want to check out the 'Blossom drop FAQ' here as there are other causes besides air temps (like too much N fertilizer) but air temps is the most common cause.

Dave