Some people give us domain registration and hosting orders by using another person's(for example his father or friend or business partner) credit cart via Paypal. We generally cancel these orders and refund moneys. But we lose many orders/customers because of this. What do you do at a similar happen?

That's a tough call. Are you able to receive a signed payment authorization form from the 3rd-party credit card holder?
-mike

I manually approve the orders by phoning the number listed with the CC or PayPal account.

A phone verification will be good. If you can afford, go for a fraud verification company.

I see some companies ask the customer to fax or email the images of front and back of the credit card with signed letter for accepting credit card. I think on that way how can do the hosting business.
What is the correct remedy for this situation? We all facing this situation tremendously.

It takes a few moments out of your day to give the number on the order a call to verify with the card holder. If they are not open or co-operate with the verification process, then at that point I would refund/cancel the order.