AMD Mantle and GPU market implications

1. Basically, better performance or the ability to extract better visuals at the same performance we have now. I'l probably butcher this explanation, but close enough.
The way it works with DirectX is that the CPU has to generate DrawCalls, which are basically instructions to the GPU to do something. Since DirectX works on all different architectures across AMD/Nvidia, these drawcalls are fairly complicated as it needs to be compatible, and this puts more strain on the CPU. The idea is that with a lower level API (how the OS interacts with the graphics card) you dont need complicated drawcalls to send requests to the GPU and they intend to parallel the computing of drawcalls, so you can use multiple cores to generate more of them. This will simultaneously reduce the CPU's impact on gaming performance and increase the number of drawcalls possible, which means the GPU can properly use all its resources.

2. Long term advantages AMD might have over Nvidia...
Their work on Heterogeneous System Architecture could be a big thing going into the future, and were seeing that starting to happen on APU's. Though this is largely because AMD are in the CPU space, while Nvidia aren't except for the Tegra chips, which are focused in the mobile/handheld market.

Mantle itself I don't think will be that long term, as it only works on the GCN architecture (so far) so it will need to replaced or updated when the next-gen GPU's come out. Nvidia could come up with their own Mantle alternative for their system architecture, I have little doubt that something is in the works along these lines but nothing is known.

You could say that AMD own fabrication plants, whether that means anything in the long run, who knows.

3. We dont know, Mantle was supposed to become available on BF4 by now, but DICE have been too busy bug-fixing to integrate it yet. Until then, we have no idea if Mantle is as good as they say.
I wouldnt say Nvidia are behind because of Mantle, they have plenty of their own stuff (ShadowPlay and G-Sync notably) that offer compelling reasons to consider them over AMD. That and they have reliably been topping AMD cards, with AMD mainly competing through superior value, not outright performance or stellar design (290/x case in point)