In this sentence correct? The couple is taking their sweet time.
Yes, it's common to think of such collective words, eg couple . . . team . . . in terms of the people involved. So, saying the above is a lot more common than saying something like ' . . . its sweet time', which sounds pretty strange for a couple.
No it isn't correct. You cannot use the singular 'is' and the plural 'their' in the same sentence referring to the same couple. Whether a collective noun (like 'couple') is treated as plural or singular depends on whether you are thinking of it as a single unit or two separate people. In this case "The couple are taking their sweet time" would be preferable. "The couple is taking its sweet time"