Product Owner - how to start role

The first thing to understand is what makes you the Product Owner, rather than somebody else. Why are you the person who owns the product? 

A few thoughts come to mind:

1. Stakeholders - Who are they?  How are they involved in your product (directly, indirectly, upstream, downstream, etc...)? Who is impacted by your product most?  What is the dynamic between your stakeholders (amiable, adversarial, siloed,  etc..)?  How do your stakeholders view your product as you begin your role?

2. Customers - Who uses your product?  What do they value?  Do the stakeholders accurately interpret what the users value? Who are your advocates and why?  Who are your detractors and why?  How do your customers view your product as you begin your role?

3. Value - How does your product produce value to the business?  What elements of the business strategy and measurable KPIs does your product touch?  What has made your product successful so far?  What pain has your product caused so far?  What commitments were made by your predecessor and do they align with what you have learned from all of the questions before this one?

Thanks for your replies this is helpful. How would one translate these questions and points into specific tasks to do in the first few days?