For example:
On the website you can buy altcoins with BTC or Eth. Would it be possible for customers to buy them instantly? I guess no, because it takes time for Btc or Eth transactions to go through blockchain and get confirmed. We can't release our altcoins if we don't get a payment confirmation.
Is there any way around? For faster transaction confirmations?
when you buy/sell inside an exchange, it's like doing it outside the blockchain, in a state channel. Every blockchain has its own processing time. There is some chain which is really fast though, such as Ripple and Iota, whose processing time is in the range of few seconds.