Cycling – I am out of shape, and to reach respectable average speed, I was gasping for air on every little hill. Should I be trying to force myself to inhale through the nose?

Yes, the best would be switch totally away from the “winning attitude” and shift gradually to the lowest gear instead. Slowly but steady uphill – with closed nose, definitely. If you can not do it, then “bite you pride with your strong foreteeth” and walk uphill – with closed mouth. After a week of doing so you will see a HUGE development in your aerobic capacity!