Well the only low powered device capable of doing all the above would be a laptop. Or you could just downgrade your components - low power Intel dual core, low TDP GPU - and save power on the Desktop itself.
The only way to measure power consumption reliably is connecting a kill-a-watt. Some APC UPSs with PC interfaces can display the rough power consumption too. Your GPU might show realtime power consumption via apps like GPU-z.
Coming to router, I am thinking of getting a most basic one with USB port. Right now I have a TP-Link 741N which I need to send for RMA as it can not be reset-ed. I am open to a router with two WAN ports if there is one in ₹2.5-3k range.
The cheapest dual WAN port router I know of is from TP Link - SOHO series. You should be able to get it for about 4-5K.