Yes. The rash could be physical irritation from the carpet. It could also be a symptom of eczema or contact dermatitis. The first thing to do is apply some thick ointment to the rash and cover with long pants. If it is itchy, you could try some Hydrocortisone cream. If it is painful or oozing, this might require medical attention.
Yes. Some skin conditions such as eczema can be worsened by the friction of rubbing against a carpet (just as scratching can make eczema worse) protective clothing or moisturizers such as keri or curel may help. If not, a doctor may recommend a medicated ointment.