Are there any cloud services that provide categorization to natural language questions via API?

Are there any cloud services that provide categorization to natural language questions via API?

Yes, there are many.  This is usually referred to as named-entity recognition, one of the common tasks of natural language processing, and works for any sentence or phrase, not just questions.

Alchemyapi.com offers a number of related APIs, including:

• Concept Tagging
• Entity Extraction

Mashape.com API marketplace also includes a variety of options:

• Entities API Documentation
• Entity Extraction API Documentation
• Noun Entity Extraction – Noun Phrase – Part Of Speech Tagger ALPHA API Documentation