Am I allowed to do this?
 

I want to go from HKG to DEL and MUM back to HKG.

If I fly this exact route, its almost 2,000USD. However, If I base out of Taipei (TPE--HKG--DEL, MUM--HKG--TPE), the flight is only 700USD. Is it possible to buy the flight out of Taipei but not actually be on that leg of the flight and just check in at the check in counter in HKG for the HKG leg? Will they give me any issues at the check in desk?

https://www.dropbox.com/s/cnkwl2xy7w...%20HK.jpg?dl=0

https://www.dropbox.com/s/psdbkqfuqa...20TPE.jpg?dl=0

If you don't fly any leg of your ticket the rest of the ticket will be cancelled. So no it's not possible.

But if you get a cheap ticket to TPE (<$1300, anyway) to fly the whole route outbound, you could just get off the plane in HKG on the return and drop the last segment to TPE. Though technically you aren't supposed to, obviously.

Not possible. If you don't board the first leg (TPE-HKG) the rest of your itinerary is cancelled.

Go to a TA in HKG and you should be able to get it for much cheaper. Search 'nanda' in this forum to see why HKG-India is so expensive when you book online yourself.

If finding a local TA is not an option do what was suggested above - buy yourself a one-way HKG-TPE.

What on earth is "MUM"? After googling, it's an airport in Kenya? Do you mean BOM or are you actually going to MUM?

Looks like OP though MUM is BOM as it is in Mumbai?

I think you got the pricing from CX web right. Buy your ticket through a local travel agent and you will get a much better price.