wondering if this is what is affecting my electric bill and my other computer being so slow
Does a gaming PC *usually* draw more power than a slower PC? Depends. Years ago, you could be rocking a core2Quad with GTX 480's in SLI...and sucking a huge amount of power; compared to a much more efficient 4670 and gtx 770 SLI.
If both machines are roughly in the same generation, then the gaming PC will usually draw more power given it's more demanding hardware (which is going to include a graphics card, which most "slow" computers wont have).
To your mbps (assuming you mean internet usage), usually not...a gaming pc's ethernet port is nothing special usually. Most services can max the NIC capabilities of older hardware.