Passing the store (or in general, your application state) down from each component to their children has one big flaw: It reduces your components' reusability in other projects.
You can circumvent this by creating a component at the top level of your application which holds the state and passes it down via React's context. The state-dependent components can then pull the state from context where they need it. By doing so, you basically reinvented react-redux, which works just like this (+ the performance optimizations).
Regarding your issue with the complete app rerendering on a single keystroke: This is a result of you passing down the entire store to each component. Since you still edit the store in an immutable way, on every keystroke you create a new store object and React rerenders everything because the store prop changed. I'm also confused how using paging/filtering would help with this.
Another advice would be to not store form state in redux. Keep your redux store only for your actual model application state. Keep forms/temporary state in your components and move it to your model redux store once it is done/baked.