60 Amp breaker Going to 100 Amp

I have a 100 amp sub panel with a 60 amp square D breaker, that leads to the detached garage with another 100 amp panel. That 60 Amp should be a 100 amp correct?

It is not violate anything for a main breaker to be of less amperage than the max in the box. If it is used for lights, it will probably be okay.

Nope. The breaker protects the wire leading to that subpanel. The “100 amp” rating of the subpanel is the max rating that you can feed it at. In other words, it’s “heavy duty-ness”. This is the same as saying that you may only carry a sack of groceries in a 3/4 ton pickup truck. You don’t need to carry 3/4 ton loads in it just because it’s rated that heavy.

Not necessarily. It can be, if the conductors are sized appropriately, but there is no rule that says it must be.