Do you lose anything by not doing an action 24/7? For example, if I finish improving relations with Italy and don't immediately start another diplo action, am I missing out or does doing that stuff cost me something? If you miss out then you should add an option to autopause when an action finishes or make those actions use up points so they can be stored up like research.

Right now you're missing out, I've felt this pain too in games at high speed. Next update will have most missions and improve relations auto-repeat until you cancel them or pick another action, and up to 180 days worth of actions will be stored to make your next action run twice as fast until they run out.