If I ask a question privately, ie. restricting it to one of my teams, will it appear to others? Will it be searchable in the archives?

If you are a member, you can opt to keep the term private indefinitely. If you are a registered user, you can opt to keep the term private for one month, after which time it will become public and searchable.

Important note: bank-like security is not being employed in KudoZ, even for private questions. Sensitive terms should not be posted via KudoZ, even when using the privacy options.