For your first question, any new one should have the "Sports" denomination; however the old version has a very good reputation as well.
I ... 70-200 on the D7200 is going to be about the same view as the 120-300 on the D750, so kind of defeats the purpose. I don't know that I really need the D750, as the D7200 is about the same size buffer, low light is mayby 1 stop difference, and my photos are tack sharp with the D7200. ...
I'm talking far future, but this is playing in my head as well.
Looking at DXO mark the D7200 does perform well, however there is a difference of full frame on colour and noise (ISO). I would also go for the D750 AF capabilities, but that is me.
DXO mark does not have the 70-200 VRII it seems, but seems to confirm that the VR I is less in sharpness then this 120-300 f2.8 which is obviously more expensive when new.
Practically for me the 70-200 is a very good performer and I was thinking if I would go for a 120-300mm f2.8 the 70-200mm would have to go. That seems to decide against the 120-300mm for me. Depending on your subject the 200mm with crop will do very well (for example using a D7200 mostly).
So I guess my conclusion is that the 70-200 with D7000 (my case) and D750 (when light is critical) is a good option which would not be much complemented with the 120-300mm.
If I would start over I would consider 70-200 vs 120-300 and would probably choose the 120-300 f2.8 for my photography (wildlife and motorsports). Mainly for quality for the price and probably can be better used with teleconverters (I checked a group on flickr for that).
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Edit just found:
Sigma 120-300mm f/2.8 DG OS HSM Sports
predecessor: Sigma 120-300mm f/2.8 EX DG OS HSM
predecessor: Sigma 120-300mm f/2.8 DG HSM