exhaust port of 55 gal steel drum

Is there any reason why I have to have the exhaust port on the bottom? Is it because the gasses need to pass down the sides to cool?

One of the main reasons why the exhaust is at the bottom is so that the barrel can then act as a radiator for the heat before continuing into the mass or up and out a chimney.
So no, you really don't have to have it located at the bottom if that fits your goals better. It certainly will add a lot more heat up and out the chimney and therefore increase draft as a result but if your fine with that then by all means go for it.

One final thought: Some people do both. They have the main exhaust at the bottom and a bypass near the top which can be opened and closed at will to either help start a cold stove or just don't want the radiant heat in the room as would be the case for an indoor cooking stove in summer.

Heck, what you need is a diesel space heater with a stove pipe on it and a transition at the other end to go to your batch box. That'll heat up it's innerds in no time.