If we're to contribute at a policy level, we need to understand how policy is conceived, designed, and implemented today. What do you know about policy and its workings?
It's an interesting question but my brain just can't process it - I can only think about what policy-makers need to know about design. When you say "we" who do you mean? Designers who design policy? Or designers who want to influence the way policy is developed? Or designers who want a specific policy to reflect their own position? Does one need to know about the inner workings of public institutions to influence policy or make it better? Isn't the onus on policy developers to include, involve, explain? There, now my bias is showing.
Perhaps its because I don't know any designers developing policies but I do know people developing policies, myself included. Having spent the last year working in a policy job, I still only know how we developed one specific policy, which was significantly different from how the rescinded policy was developed. It's also different from how other policies have been developed by other people in my organization. From that, I presume that every organization has their own process - and also that individuals shepherding the process along influence it throughout. Each policy also has it's own set of constraints, competing interests, political will and public interest.
I'm not sure how often the process is as intentional as the words you've used here - conceived, designed and implemented. In an ideal world, the process would start by defining what a successful policy would look like or do. Any stakeholders who would be affected by the resulting policy would know how they could contribute, because the process would be designed to facilitate involvement; taking action on feedback would be part of that intentional process.
So, to answer your first question, anyone working in policy, or wanting to influence it, might want to know:
Policy works when you know how it is applied to certain sectors.
ex. Employment policy has claimant courts and ombudsman organizations to give employees options to improve their work situations.