My denomination has
Arminian roots. And I always have to double check that I'm not saying
Arianism -- a mistake I've made before. Similar names, but very different issues.
As I understand your question, you are asking how God, who is more powerful than humans, acts in regards to prayer. It seems that your understanding is that the purpose of prayer is to persuade God to act a certain desired way.
Daniel, in the Bible, was told by the angel that he was highly regarded in heaven. If Daniel asked God for a particular favor, I suspect that either God or perhaps various guardian angels would be happy to grant Daniel's wish. This, of course, would be tempered by the feasibility of the request and what effect it would have on others etc. And this would be an example of what you wrote about, praying for unbelieving family members.
I think another aspect of prayer is changing US, rather than changing God. A silly example would be if I kept praying to God to make my wife stop throwing dishes at me. And with time, God is able to help me see that the problem was with ME rather than my wife. And once He changes my heart, suddenly the crashing dishes stop. (To be clear, my wife has never thrown any dishes).
Another aspect of prayer seems to be that of giving God permission to act. He doesn't need our permission, but God seems to respect boundaries. He doesn't force His will or wisdom on those who don't want it. But if I ASK for God to help me, He is ready to spring into action.
So in my mind, I pray because I want God to help me change so that I can be more aligned with His ways, and thus more effective. And by actively asking for His help, I hope to remove any barriers that might keep Him away due to politeness. And I also pray asking for favors, as a child might ask their father for presents.
Does that answer what you were asking?
Best wishes
KT