1) The FX-8350 will definitely bottleneck SKYRIM. That's been well tested.
2) The FX-8350 will provide a slight improvement versus an FX-6120 though it's likely just mainly the higher FREQUENCY. Note the following benchmark differences between an FX-8350, FX-8150 and an i5-3570K.
http://www.bit-tech.net/hardware/2012/11/06/amd-fx-8350-review/6
Note that even at 4.8GHz the FX-8350 isn't even close to a modern Intel CPU (and 4th gen Haswell is a bit better than the 3rd gen).
More benchmarks (The highest benchmark above has no FXAA enabled but I'm surprised that the difference would be that great):
http://techreport.com/review/23750/amd-fx-8350-processor-reviewed/5
or
http://www.tomshardware.com/reviews/fx-8350-vishera-review,3328-14.html
3) The FX-6120 with a weaker graphics card seems to bottleneck above 60FPS at 1080p on HIGH/Ultra (probably slightly higher with unofficial patches).
*Most MODS are demanding on the graphics card especially so you might still be able to maintain 60FPS while also using certain mods. Be careful though because even the best machines can be bogged down by certain graphic mods.
SUMMARY:
**So your CPU will definitely be a bottleneck in the sense that a better Intel CPU would produce a higher frame rate. However, what is most important is that you should be able to run at a solid 60FPS with a few mods running which is quite smooth so it's not really an issue for this game.
Make sure to use the Nexus Mod Manager and apply the Unofficial patches for Skyrim, Dragonborn etc. as well as use FRAPS when adding mods to ensure you maintain a solid 60FPS if possible. You can get a bit of STUTTER if you drop below this.
I think you can force ADAPTIVE VSYNC with the NVidia Control Panel which would disable VSYNC if you drop below 60FPS (for 60Hz monitor). I suggest doing that to prevent the STUTTER you get if you can't achieve 60FPS but have VSYNC enabled (it's forced on by default).