I wouldn't mind is a phrase that has a couple of meanings, depending on context
- [something] would be OK with me - I wouldn't mind if you used my car
- [something] is not OK with me (used with but) - She ate my last apple! I wouldn't mind but she doesn't even like them
- I would like [something] (expressing desire using understatement) - I wouldn't mind a holiday right now
Your examples are using the last meaning, expressing that you want something. The reason your first example works is because the [something] phrase functions as a noun, to cook does not. By changing it to a gerund phrase like cooking something to eat makes it work as a noun, and that fits the phrase.
I wouldn't mind a house to possess works fine too, because that's a noun phrase as well - it just sounds unusual. There's nothing wrong with the words (although possess could feel redundant), but these would all be more common ways to say it:
- I wouldn't mind owning a house
- I wouldn't mind having a house
- I wouldn't mind a house ("having" it would be understood in context, but this version feels a little ironic because the word is missing, it's almost passive, or implying expectation that things should just work out that way, with you owning a house.)