"1. If the UPS is sharing a GND connection with the Input connector and the output socket I can get away with single wiring right?"
Given that you are absolutely sure about Neutral being connected between Input and Output of the UPS. You may just use the Live wire from the UPS Output. And that the Input of UPS is connected to a socket providing it with the same Neutral as of the load.
"2a. Ideally I would want the UPS inside my room beside my computer and have the battery outside putting a distance of around 10 feet between the two. How much distance can I put before voltage drop becomes an issue?"
Would be better if the DC wires in between battery and UPS are kept short. Low DC volts, high DC amperes, Power = I2R; hence more power wastage.
Better to extend the wires of 220V AC output from UPS. Less losses.
If you must have 10 feet of distance between UPS and Batteries, assuming battery bank is 48V DC, at 700 watts, you are looking at about 15 amps; I would suggest using 6mm2 wires for DC; as per online calculators the drop should be around 0.7%.
"2b. If I have the UPS beside my computer how would I wire other stuff beside my computer. This includes a fan and some lights. This seems impossible so far, I am afraid I will have to put the ups and the battery near the main circuit breakers if I have to power my room."
If you can clearly identify the live output from UPS. You may remove the switch plate of your room, remove the live wire coming from your main to your switches and attach the live from your UPS to the switches.
"3. When the UPS is being powered from the mains it seems that the left hole of the output socket is Live and the right hole is GND. However during loadshedding the tester lights up on both the holes. Why would this even happen, I am stumped. The tester also lights up when I touch it to the UPS's input connector (that plugs in the wall socket to charge the batteries) and on both terminals, these terminals register 1V on a multimeter. I was worried this would put the entire room's load on the UPS, but that is unlikely with 1V. From my understanding the tester shouldnt even light up on the input connector."
Will not say much about the above as I am not fully sure about this.
Normally, when facing a socket, the right-hand side connector/hole should be live.
My sincerest advice to you is not to combine the UPS with your home original wiring. It is just 3 wires(Live, Neutral and Earth) but a lot may go wrong.
Buy an additional fan(pedestal or the one that hangs on the wall) a LED lamp(tube/bulb) and directly connect them to your UPS output as and when required. Same for the computer. Just use the output of UPS directly.
If you still want to experiment with your home wiring, then do use 2-pole breakers on both the Input and the Output of your UPS. 1400VA at 220V would need around 6 ampere breaker.