You could also use 'mind' in sentences such as 'Could you mind the children?' (meaning look after) and 'mind your language!' if someone were to swear. :)

When "mind" occurs as a verb, then an object takes the form of a noun, of course. In the example "would you mind answering?", the word "answering" is a gerund (the -ing form of a verb used as a noun). It would have to be in that form. (the verb "mind" is one that always takes the gerund as an object, rather than an infinitive.)