"Would you mind to do something?"

Is it correct to say "Would you mind to do something?". I've seen this usage in a few places, but it doesn't sound right to me. I would guess that it's proper to use "Would you mind doing something?" instead.

Yet there are other verbs we could substitute here that make it sound correct with "to". For example, "Would you like to do something?" and "Would you care to do something?"

Could someone explain why "mind to" doesn't sound correct in this context, while "like to" and "care to" sound exactly right?

No, it's not correct.

The verb mind can take an Equi Gerund Complement clause:

• Would you mind doing something?

but not an Equi Infinitive Complement clause:

• *Would you mind to do something?

whence the asterisk on the second sentence, indicating that it's ungrammatical.

Verbs vary a lot in which complement clauses (if any) they can take; this is one part of the meaning of the verb. Like and care are both verbs that can take infinitive complement clauses; mind isn't. That's all, really.

Any good ESL dictionary should indicate which types of complement clause a verb can take; dictionaries for native speakers, on the other hand, usually don't mention that. One more reason not to look in dictionaries for grammatical information.