I don't really know what it is, but if I had to guess it sounds like a grafted tree that was overtaken by a root-stock sucker. I'm sure someone else will be able to help you better.

Yes, you almost certainly have an orange tree, but the rootstock has been allowed to sprout and take over. The rootstock is rough lemon or perhaps citrumelo or citrange. Are the leaves different where the different fruits appear?
The nearby orange tree will not affect this in any way at all.
You could cut away all the rootstock growth which will originate low down beneath the original graft point, but it doesn't sound as if you'll have much left!

Thanks for telling me what to look for.

Yes, there is a scar down low(about 3" above ground) on the main trunk. Then the trunk splits into a v shape. One thick branch is the orange section. The other one splits yet again, and those two form the vast majority of the tree.

The orange section is mostly small on one side, maybe 2' by 2'. However one large branch stretches across into the lemon section and has the majority of the oranges. It's maybe a 6' by 4' section.

So while I'd lose the vast majority of the tree, it won't be totally bare. I think I might go for it. I plan to live here for a while, so I might as well make things look the way I want. Thanks again.

My Eureka lemon used to have occasional thick & lumpy skinned fruit (outdoors in SoCal.) As I recall, it was a cultural thing.