When a contact initiates a conversation via a Live Chat widget, you will see that event displayed in the Activity Tab on the Contact Detail page in Contacts.
Choosing the clipboard icon will open the form submission (aka the initial message they entered into the Chat Widget) where you can view more information including the source (Organic Search, Direct Traffic, etc.).
Alternatively, you can view the traffic source for Live Chat conversations by selecting "Widget Form" in the Form Selector drop down menu under "Sites" > "Form" > "Submissions" Tab.
Yes, attribution will work for Live Chat Widgets added to Non-Topline OS websites, including sites you use the LeadConnector WordPress Widget on.
Attribution may not work for some iOS users or privacy mode users as it blocks the cookie.