Tomato is a very strange fruit, or vegetable. Which one is it again? 
It's actually a fruit. A fruit that of course is edible and red, called the "Solanum lycopersicum".
What a confusing genus species name. Do you happen to know when it was first domesticated/harvested?
Well, the Spanish found it when they interacted with the Aztecs, when the Spanish colonies of America. This led them to bring it to Europe in the 16th century!
Interesting, are tomatoes grown off of a plant or do they grow underground?
They grow off the plant, as a berry style. They are really thought of as a culinary vegetable, the perfect ingredient for savory meals and sides!
I love tomatoes for the most part, but I cannot stand ketchup!!! (I call it tomato blood)
There are many people like that! Tomatoes are so good in many ways though. From eating raw, to adding to salads, sauces, dishes and even drinks, they're very handy!