Can I use a 20 amp breaker in the situation of having 15 amp wire connected to a 20 amp main wire?
No, in a situation like this absolutely not. The breaker in this case would need to be 15A. 

Question is, WHY? Why use #14 wire when the circuit is already #12? Stick with what is there and keep the breaker 20A. In some cases, such as in a kitchen, laundry or bath, you'd be creating a violation doing it like you show in that diagram.

And don't be confused by the receptacles being "15A rated". They are FINE on a 20A circuit and are 20A feed-thru rated as well.