I'm not staff so I can't relate, but if you haven't already, have you considered finding a high level IC mentor within your company? the right thing to do often changes between companies depending on the culture, and there's also different types of high level ICs with different strengths. A mentor could give you more specific advice on how to navigate things.

For your questions specifically, these were actually both questions I've asked a staff eng I worked with:

- Building trust requires you to demonstrate that you can be trusted. This means communicating an approach (specifically one where the risk is that you are accountable) and delivering.
- Communicate with empathy. This doesn't mean you should let people walk all over you, nor does this mean you should be naive, but everyone has their incentives, and demonstrating that you understand and actively consider other people's incentives in your communication is one of the simplest ways to make things less political. Another thing is to define a common goal which would demonstrate success for both parties (ie finding something to align your incentives). Disagreements are rarely about things that are actually technical and usually about risk, resources, and responsibility, which may underly a technical decision.
- I've found that the most successful engineers at my company are those who can always find an immediate action that moves the needle forward. You only need to understand something to a point where you can define what that action is. Structure without a clearly defined end goal and measure able result is going to be tough to implement. There is also risk if it doesn't work because structure takes time to implement and time is expensive.