Yes, you should if you are referring to a specific box. For example, "This is the Lost and Found box." However, if you are referring to the concept of a lost and found box, then you do not need to capitalize. For example, "We should make a lost and found box for people to use.

No. Generally you would only capitalize nouns that start a sentence or are "proper nouns", i.e. named people, places or things. I don't imagine your box has that status, does it?