Probably the kitty is feeling a little under the weather. Make sure your cat is eating and going to the bathroom like usual. If your cat uses a litter box and you can watch, see if kitty cries while going potty or if there is any straining. If any if these things are evident I would call Vet and explain symptoms. Better safe then sorry. Good luck.
Yes, cool and dry or cool and slightly damp is normal for a cat nose. However, if it's been sleeping curled up (nose tucked under) the nose might be warm and dry for a bit and be ok. However, if the warm/dry nose goes with a sluggish or disinterested behavior I'd be _very_ concerned. Cats cover up illness well, unfortunately. There are other general health warning signs to watch for, like decreased appetite, drinking more (or less), runny stool, difficulty peeing or pooping, peeing or pooping away from the litter box, hot ears (feel the upper part for excessive warmth), runny eyes or nose. Any of these signs, alone, would warrant a vet visit. But sometimes the only sign of an internal problem (or infection) may be a continuously or off-and-on warm nose or ears. Cats can quickly succumb to things like urinary tract infections (very curable if caught early) with very few signs until they collapse. I would say a visit to the vet is called for, to be sure.
A healthy cat's nose can vary between wet and dry several times over the course of a day. And there are many reasons your cat can have a dry, warm nose that have nothing to do with health. Here are a few: Lying in the sun Spending time near a source of heat Dwelling in a room with poor air circulation Licking — saliva quickly dries on skin Looking at a cat's nose is a good way to check for some other problems. If your cat's nasal skin is flaking, a dermatological problem may be to blame. Ask your veterinarian to check it out. When examining your cat's nose, another thing to look for is nasal discharge. If your cat's nose runs, the mucus should be clear. If your cat is producing bubbly, thick, yellow, green or even black mucus, see your veterinarian.