In the poll above right, all of the following are correct, except for
*Would you mind to explain that again?

Any verb directly following ‘mind’ takes the -ing form (“Would you mind explaining that?”), not the infinitive. Two other verbs that belong to this group include ‘avoid’ (“avoid mentioning her name in his presence” and ‘suggest’ (“we suggested taking the train, but they insisted on driving.”)

Further examples include:

I hope you don’t mind the noise.
Do you mind if I open the window?
Would you mind explaining that again?
I don’t mind helping if you can’t find anyone else.
Please read that contract over again, if you wouldn’t mind, before sending it.
I wouldn’t mind a cup of coffee, if it’s no trouble.