Modern motherboards do support pcie 3.0, including yours.
Check the authoritative spec on the gigabyte web site.
It is a moot point since pcie2/3 are both forward and backwards compatible.
Performance difference is negligible for all except the strongest of cards.
The minimum recommendations do suggest a 7870 card which happens to have 2gb of vram.
The amount of vram is a performance factor not so much a functional thing.
A game needs to have most of the data in vram that it uses most of the time.
Somewhat like real ram.
If a game needs something not in vram, it needs to get it across the pcie boundary
hopefully from real ram and hopefully not from a hard drive.
It is not informative to know to what level the available vram is filled.
Possibly much of what is there is not needed.
What is not known is the rate of vram exchange.
Vram is managed by the Graphics card driver, and by the game. There may be differences in effectiveness between amd and nvidia cards.
And differences between games.
Now, it is possible that a game is so poorly coded that it will not function at all with insufficient vram.
Regardless, the 7870 is a stronger card than yours, hence the suggestion to upgrade. GTX1050 is a modern card that is slightly stronger than the 7850.
Tom's GPU hierarchy chart will give you the relative strengths of cards:
https://www.tomshardware.com/reviews/gpu-hierarchy,4388.html
If you want three monitor support, check on what outputs your monitors can accept and match that up to what the graphics card can output.
GTX1050 will have a dvi, dp and hdmi output.
Adapters may be necessary if the outputs do not match.