If I am using an online broker, Merrill Edge, is it possible for my transaction to settle before the third day? I am confused on the settlement process, in theory, couldn't I settle the funds immediately? Why would I then have to wait for T + 3 to trade (specifically sell) another security with those funds?
It is possible but unlikely.
Securities firms would prefer never to settle externally; rather, they prefer to wait until the liabilities can be netted. They are forced to make and take payment in three business days.
The reason why is because settlement is costly in the same way as any other business would prefer to build trade credit instead of taking or making payment rapidly.
The only circumstance where a financial firm would wish to take full delivery is when a counterparty is no longer trusted to be solvent.
The T+3 settlement date only affects cash accounts. In a cash account, you need to wait until the T+3 settlement date for your funds to be available to make your next trade. But if you convert your cash account into a margin account, then you do not need to wait until the T+3 settlement date for your next trade - your broker will allow you to make another trade immediately.