[SalesForce] Button to Open a LWC

URL addressable tabs are currently not supported for LWC components as documented in Unsupported Experiences and Tools

Also from Documentation:

To make an addressable Lightning web component, embed it in an Aura component that implements the lightning:isUrlAddressable interface.

So embed LWC component in Aura component and have navigation to that component - using isUrlAddressable interface.

Also, pls note that the URL parameters are accessable in LWC component directly. You need not pass them from Aura component.

import { LightningElement, track, wire } from 'lwc';
import { CurrentPageReference, NavigationMixin } from 'lightning/navigation';

export default class Poc extends NavigationMixin(LightningElement) {
    @wire(CurrentPageReference)
    setCurrentPageReference(currentPageReference) {
        this.currentPageReference = currentPageReference;
    }
    connectedCallback(){
        console.log("URL Parameters => ", this.currentPageReference.state);
    }
}

As shown above, use wire service to get the URL parameters (as state) in LWC component

If you look at the documentation, to programmatically register your event listener you need to do the following in your "parent" LWC (an ancestor of the LWC generating the show toast event):

constructor() {
    super();

    this.template.addEventListener('lightning__showtoast', this.handleToastEvent.bind(this));
}

I tried this in a playground but the playground doesn't appear to correctly support toasts. It therefore looks like an exercise for the OP to see if this works.

I would assume you would need to provide a component that wraps all the others (perhaps via a "slot") in the page in order to catch "standard" toasts from these components, but I don't think you can catch toasts from the "chrome" of the page outside the body content.