Why did they disable it? Was it affecting the network somehow?
No, it was completely useless. You wouldn't even earn a cent using your CPU on the mainnet nowadays.
I had read that in order for a transaction to be counted on all blockchains in the world, it is required to exist at least 6 confirmations.
For small payments, even 1 confirmation is enough. Confirmed blocks are propagated quite fast (orphaned blocks are still a thing, though). The amount of confirmations ensures that coins won't be double-spent. Some wallets and services consider 3 confirmations as safe enough.
But if I am alone on my local network, how will I confirm my transactions?
You confirm transactions by including them in blocks and mining them. The first confirmation is when a transaction is included in a block. Each new mined block adds another confirmation.