A good backup strategy should include periodic recoveries to ensure that the backups are useable.
That being said, you could print out (or write down) your private keys on a regular basis. Then you could just look at the paper with your eyes and see if it is "damaged". If it is faded, and you can no longer read the ink, or if the paper burns up in a fire, or if a piece of the paper including some portion of a private key is torn off, then the "backup" is "damaged".
If you have a large sum of bitcoin that you are concerned about, you could also consider printing out a paper wallet and sending a large percentage of your bitcoin to that paper wallet, keeping just a small percentage in your digital wallet for your daily use. This way if you discover that the backup of your digital wallet is corrupt (or stolen), you will only risk losing this small portion of your holdings.
1. I make regular backups of all important files, but how do I verify that my wallet.dat file is not damaged?
Try recovering it somewhere (another computer?)
2. What if my hard drive has bad sectors and can it affect the wallet.dat file?
Depending on how many bad sectors and where they are located, yes it is possible.
3. Will the wallet.dat file still open even if it has like a few bytes missing?
I learned something new here in Mr. Gornick's post above. Apparently it will?! That's nice to know. I hadn't realized that before this.
4. Could I be backing up a damaged file and not being aware of it?
Absolutely. It is possible. As long as your wallet successfully runs, the odds aren't high, but it isn't impossible.
5. What if my computer crashes or there is a power failure while the wallet.dat file is being modified by the bitcoin software?
In most cases you will be ok, but a good backup strategy will certainly reduce the risk.