Viewport, I believe it's called
 

Is there some way that you can make your toon avatar not in the center of your screen, but more towards the bottom? This way, there's more screen room in front of your character, when the action generally is.

Yes, it is called a viewport. There are quite a few addons that can do this (some only the viewport part, others more). Since I forget exactly how to link an addon search and have it keep the search results (instead of go to the search page), click on the "Find" tab above. Check "title" and "description" and search for the word viewport. :)

This is not possible. The only time the camera switches anchor- or center point is when you mount or dismount from a vehicle.

You will not be able to change the camera to see more ahead with viewport addons. Think of them as changing the resolution. Like if you change top and bottom it is like using a widescreen resolution and you will see more on the sides but if you change right and left nothing really happens except for hiding the world.

The only way to see more ahead of you is to turn the camera, or rather tilt it so you're closer to the ground and see more of your back.

He doesn't ask about seeing more, but simply about making the important parts 'bigger' on his screen, if I understood correctly.

Don't think so. He wants his character in the lower part of the viewport (!), to have more room "in front" of the character. This isn't possible though. Changing the viewport just enables you to use a smaller resolution for the actual ingame stuff, so you can place interface elements around it without wasting that space.

Ah yes, I guess I skimmed over the rest of the OP.

What you are specifically asking for (at least what I and v6o *think* you are asking for) is not possible.