[SalesForce] When do @wire methods run (LWC)

Going by the documentation, it seems you will need to create an additional property and utilize that in your wired function.

To refresh a wired method, pass the argument the wired method receives (which is the wired value) to refreshApex(). In this sample code, the wired method is wiredGetActivityHistory(value). Hold on to the value provisioned by the wire service and pass it to refreshApex().

You will need to declare another variable, say mySendForSplitting and then utilize that in your sendForSplitting, something as:

sendForSplitting({error,data}) {
    this.mySendForSplitting = data;
    // rest of the code
}

And then use refreshApex() as below:

refreshApex(this.mySendForSplitting);

You can find an example on the lwc-recipe as well.

1. Wire service caching depends on the apex method parameters. For same parameters' values, the same apex method invoked from multiple components will have only 1 apex call.

2. For same apex method but different parameters values, there will be a separate server call to get response.

It will be better understood when explained with example. Consider a parent component:

HTML:

<template>
    <div>POC:</div>
    <lightning-button label="SHOW" onclick={showcomps}></lightning-button>
    <template if:true={show}>
        <c-child1></c-child1>
        <c-child2></c-child2>
    </template>
</template>

JS:

@track show = false;

showcomps() {
    this.show = !this.show;
}

When you render the components by clicking on button, here will be only 1 apex server call:

child1.JS:

@wire(searchAccounts, { str: 'test' })
search(result) {
    console.log('1 result => ', JSON.stringify(result));
}

child2.JS:

@wire(searchAccounts, { str: 'test' })
search(result) {
    console.log('2 result => ', JSON.stringify(result));
}

Here will be 2 server calls: (Since the str value is different although same apex method)

child1.JS:

@wire(searchAccounts, { str: 'united' })
search(result) {
    console.log('result => ', JSON.stringify(result));
}

child.JS:

@wire(searchAccounts, { str: 'test' })
search(result) {
    console.log('result => ', JSON.stringify(result));
}

Additional question: If I have multiple components importing the user ID import USER_ID from '@salesforce/user/Id'; is that being cached after the first component imports it and then the value gets imported from the client-side for the rest of the components?

There is no concept of caching for imports.

Imports are retrieved directly along with components. There will be no separate server calls. This can be verified using same above approach.