Can the Province (unbroken) abilities be used at any time an action can be used? Or only when the province is involved in an active conflict?

Any time an action can be used. It's just most specify " Action : During a conflict at this province - " thus limiting them to only being usable when the province is involved in an active conflict.

A good example is along the river of gold, which can be used during any conflict as long as it happens in a water province.

Or Magistrate Station which can be done at any time, including outside of conflicts.

The only inherent restrictions on actions and abilities you can do at any time are:

1) Cannot play (buy) characters from a province outside of the Dynasty phase

2) Cannot Attach attachments during the dynasty phase

3) Cannot play conflict characters from your hand during the dynasty phase.

All other abilities/actions are valid and apply unless they say otherwise, such as "During a conflict".