I've seen a lot of users who are high in reputation but have never asked a single question. Are there any requirements, like "You have to have asked at least 1 question to ____"? Should there be? I've seen a mix of satisfactory and unsatisfactory answers from these users, so I guess it's just partly my view that people who answer questions should have asked one.
No, there are no such requirements and there shouldn't be any such requirements
Everyone is allowed to do how they please as long as they stay in line with the community guidelines. You are a lurker that's mainly here to upvote funny stuff? Great for you. You like editing? Perfect. You only ask questions? Welcome aboard. You only ever answer? Nice.
Everyone's allowed to do as they see fit. Some things give you reputation, like answering and asking stuff (or editing until you are at 2k rep). This is what the overall StackExchange network views as especially useful - we are here to build a database of questions and answers after all. These things allow you to do other things, like seeing deleted posts or casting close and reopen votes. But other than the normal privileges and the generic rep requirements for them there shouldn't be any restrictions, apart from the special abilities elected mods have, like seeing flags on comments that should be deleted.
Not asking questions is fine.
For more information about my view on this topic you might want to check out my answer to What does being a 'Contributor' look like on Worldbuilding?