Oh, my goodness, you didn't have to quote those long scriptures. Let's deal with them one at a time.

James 4.1-17. I presume you're referring most specifically to the last couple of verses 13-17)? The point of the text is that worldliness can cause people to neglect God and to run their lives as if He doesn't exist. Instead, we should seek the Lord for His input, considering what the Lord would have us do. That doesn't mean He has our lives planned down to every detail. It means rather that, as Garry Friesen writes (Decision Making and the Will of God), "James commands that believers adopt a humble attitude in planning," an attitude that properly recognizes the place of God in our lives.

Jeremiah 29.11-14. This is a specific promise given to the nation of Israel pertaining to a particular situation and promise. It is not addressed to individuals. It is not a promise for us. It is not a message about any plans God has for us in our lives. It pertains to Israel's reinstatement in the land. God will bless them, allow them to prosper and give them a future there.
The message for us is that God punishes sin but doesn't desert us there. God disciplines, but God also remembers; God punishes, but doesn't punish forever (Lam. 4.22).

Romans 8.26-39. I don't know why you've brought this into the conversation. It doesn't say anything about God having an individual plan for each one of us. God's sovereign plan for us is to be conformed to the image of His Son. You'll have to explain why you brought this passage into the conversation.

What about James 4.1-17, Jeremiah 29.11-14, and Romans 8.26-39?

I don't see in Scripture where God has "a plan for everyone." I don't see where our lives are either completely mapped out or even partially mapped out. I see specific instances that are planned, but nothing more.

Instead, life is in flux, there are bazillions of cause-and-effect complexities happening all around us, people use their free will to make their choices, other people's choices affect us sometimes whether we choose or not. Sometimes we are the effect of their choices, sometimes the victims, and sometimes we can use our choices to change things.

I don't see in the Bible, except a few times, where someone's sickness is God's plan. I don't see, except for a few times, where when someone dies that's God's plan.

The point of prayer is to communicate with God. Most of the prayers in the Bible pertain to our inner, spiritual lives, not to our external circumstances. And when they do pertain to our external circumstances, most of the time they don't get answered (at least not positively).

Specifically, why do we pray for help when people are sick/injured or going through trouble?

For a bit of context, I was raised Christian and my parents are still pretty religious. My beliefs have changed a lot since then and sometimes I'm not really sure what I believe anymore (picking a flair here was super difficult). Sometimes this leads to some friendly discussions when I visit my folks, and this question came up the last time I was home. The way I was raised, it was understood that God had a plan for everyone. So if someone gets sick, that's part of God's plan. If they die, that was part of God's plan. If they recover, that also was part of God's plan. So in the end, what is the point of praying to ask for help? Either the plan is for them to get better or it isn't. Praying shouldn't have any impact on that, right? And if it does have an impact on it, that raises a ton of further questions I have.