The male line carolingian dynasty is gone, it has been for around a thousand years. If you are asking about claims in the female line, absolutely! Almost too many to count. The Capetian Kings of France who succeeded the Carolingians traced female descent from Charlemagne. Their cadet branch, the Bourbons are still in power in Spain and Luxembourg (albeit as figurehead monarchs). More importantly, almost every European royal family has married into this family at some point, so if we use this as the standard every royal family has some descent from Charlemagne.

What do you mean by 'direct' ? Because if you go back far enough e.g. to Charlemagne then mathematically speaking (or so claims statistician Joseph T. Chang) every human of European descent is related to him. There are however no descendants in the direct male line.

Yes. I speak only for the English monarchy, though I'm sure some other modern rulers can trace their lineage back to the Carolingians as well.

Queen Elizabeth II, the current reigning monarch in England, is the 22nd generation great grand-daughter of William I King of England.

William I's great great grandmother was Adele of Meaux. Adele was the 4th generation great grand-daughter of Charlemagne.

This makes Queen Elizabeth II the 33rd generation great grand-daughter of Charlemagne.