How do cookie consent banners work on the background? Can a website that sets 3rd party cookies on another website ask to allow its cookies?

I'm working on a hobby project. The project is basically an integrable live support service. To describe my questions easily, I will call my service service.com and call the website that uses my service website.com. I'm thinking on implementing session management to restore disconnected visitors chat. To do that I'm planning to use cookie based session management. If owner of the website.com wants to use my service I will provide them a JavaScript file which will inject some HTML on the body, style tags on head and implement interaction. All the website.com's have to do will be importing that JS file and calling a function defined by that JS file. To set 3rd party cookies on that website.com from my service.com I will use this request/response. When website.com requests my JS file from service.com, my service will respond the request with the JS file along with a cookie to manage visitor's sessions. This way service.com will set 3rd party on website.com's visitors.

1st Question: Could this stage of setting cookie on website.com's visitor done on the front-end with that requested JS file or locally (from the website.com's web server) requested JS file? Would that still be a 3rd party cookie since it would be set on the front-end of the website.com?

2nd Questios: My other question is about cookie consents. Can a website that sets 3rd party cookies (e.g service.com) on some other website (e.g website.com) ask to allow their cookies on that website.com? In other words, can I ask website.com's visitors to allow only 3rd party cookies that are set by service.com with the JS file I serve/give to website.com? Would that be legal?

3rd Question: How do cookie consent banners work behind the scenes? What happens when you accept/deny all of the 3rd party cookies used on a website? Or what happens when you filter and accepy only a few of them? How does the process of allowing/disallowing work? Is there some kind of JavaScript that is triggered when you click that "Accept" button or "Decline" button? You can provide me any resources on this topic.

Thanks!

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