One can apply for refinancing at any time after your purchase. Normally, it is good to refinance when the interest rates have dropped or the credit score of the borrower improved.
It is not a good idea to refinance the auto loan when the current loan is going to expire soon. It is usually good to pay off the current loan when there are a few payments left on the loan. The new loan may extend the loan term and cost more with additional fees and interest.
How much one can save depends on several factors including the current interest rate, loan amount, length of the loan term, the credit score of the borrower, etc. Normally the borrower can save on monthly payments with a lower interest rate.
It is possible to refinance with bad credit. But it may not work in favor of the borrower as it may result in a higher interest rate and higher cost. It is better to check with the loan consultant for any opt.
Any refinancing would require a hard credit inquiry. This will cause a decrease in credit score slightly. But it should recover with on-time payments and keeping balances in check.