Yes, a dog will sometimes lower its tail when ill, but a lowered tail does not necessarily indicate sickness. Most often a lowered tail is indicitive of the emotional state of a dog. A happy dog has a raised tail.
Yes. You should absolutely be aware that your dog may be sick if the tail is down. A sagging of the tail signals non-alertness and a lack of energy. Also, if the dog is in pain, he or she will naturally hold the tail closer to the body for comfort.
Learning to read your dog's body language also involves paying attention to its tail, yes. When my dogs have been stressed or ill, they walk around with their tails down.