question of Clemens Cap

A propos question 1, there is always the example of a Scott monoid which is a CCC with one object D such that (necessarily) D = D x D = D ==> D. Ok, it is not quite a CCC, since it lacks an empty product, but add one so you have two objects 1 and D with the above properties. This answers question 2 too, but in addition, Beck showed over 20 years ago that any monoidal category with a coherent associativity isomorphism is equivalent, as a monoidal category, to a strictly associative one. Michael Barr