Recommended set up for Individuals -> Member of Organisation -> State Governing Body
It took me a minute to get my head around this because I'm not sure what a "band" is in your context - I haven't seen the word used that way before.  However, I realized that I could answer your questions regardless.

Question 1:  Bands should be recorded as organizations - possibly as an organization subtype, but if they're the primary type of organization you track, that may be unnecessary.  The reason to track them as organizations is...

Question 2:  This allows you to use CiviMember to track memberships for individuals based on whether the band is in good standing.  You're correct to create a relationship between individual and band.  Now - when defining membership types (Administer menu > CiviMember > Membership Types), you must set the "Relationship Type" to match the relationship type you used to connect the individuals to the bands.  This will automatically extend a band's membership to an individual.

Question 3: I would actually argue that if you need to track both type and date, a new activity type "Award" is most appropriate.  Groups and tags won't let you record the date the award was given.  Custom fields would, but you could only issue one award per individual unless you used multi-record custom fields, which are awkward and best avoided.  However, activities actually fits quite nicely and will report well.