I don't think you can search user id on frontend. There's no actual user id field so it has no search capability. The query field is also a calculated field and nothing is stored to database so it can't be searched. You'll likely need to create a new fieldtype plugin that displays user id and searches user id.

Sure you could use the Field action to set a field (probably integer) to [user_id]. What trigger you use is up to you however, but best is after registration so all future users automatically have the field populated. You'd need to manually fix existing users though or have it set on login.