Your router appears to have 4 Gigabit LAN ports so it's not really slowing down your new Gigabit switch at all.
I used the gamer lounge router long ago but I now use the D-Link DIR-655 which has wireless also.
I went with Cat6 awhile back also.
The general difference between category 5e and category 6 is in the transmission performance, and extension of the available bandwidth from 100 MHz for category 5e to 200 MHz for category 6. This includes better insertion loss, near end crosstalk (NEXT), return loss, and equal level far end crosstalk (ELFEXT). These improvements provide a higher signal-to-noise ratio, allowing higher reliability for current applications and higher data rates for future applications.