Late Payments?

Do payments that are only a few days late harm your credit rating or can they prove to be totally destructive to a credit rating even if the tardiness is minimal?

The minimalness of the tardy is not relative. When a payment is late, it is late and reported as such. If it is less than 30 days late, it will be reported as such. If it is 35 days late, it will be reported as such.

As far as it devastating your score, the answer is no, not initially. But if you make a habit of being late, your score will take a hit. If I was looking at your credit report and saw a string of less than 30 day lates, I'd wonder why you didn't request a new due date from your credit card issuers. This would most assuredly, assuming you had the money to pay, solve your late scenario.