Most minnows I know of are cold water, and although they can survive warmer temperatures it is suggested to keep them in their own tank, without bettas
However, I have only ever kept one betta with danios (leopard danios), and I made sure he had hidey holes and there were plants. They did get along, which was nice. Although, he was raised with fish on purpose (by me) and he'd always been gentle. Only time he was rough was when he had to headbutt them out of the way for food - that's the problem with fast fish...they eat first, and they eat everything
I agree with everyone about cories. Cories are not aggressive to bettas (from what I've found), and if you supply enough cover and hidey holes, bettas will not bother them