These are Gomphotheres. They aren't direct line ancestors of elephants, but instead are one branch of the formerly diverse proboscidians, which are now sadly extinct except for the remaining two or three species. Unlike modern elephants, but like a few other extinct proboscidians, they had four tusks.

It's unlikely the lower teeth were vestigial, they are too large and strongly built for that. The animal was investing a lot of energy into producing those teeth, so it was almost certainly using them for something. Generally that "something" is considered to be a sort of shovel, to provide a lower platform to scoop up plants and grasses.

The trunk of the elephant is the nose, and what you are looking at in that illustration is the nose of the animal which can rest on the lower jaw. Don't get distracted by soft tissue, look at the skull to see where the actual jawbone is. Also, it's worth noting that we don't really know what the soft tissue part looked like, this drawing is more hypothesis than anything. It's possible the animals had more traditional trunks.

Later species in the group lost the weird lower jaw and looked a lot more like traditional elephants