Hi, I'm not totally sure what you are looking for so you might want to elaborate on what you want exactly if the following answer doesn't really cover it.
By notifications, I'm guessing you mean e-mails. There is no built-in thing in Shopify for Partners to use. You'll have to integrate with an email service for every trigger that you register. I can recommend a service like SendGrid. They have a generous free plan and their paid plans are very fairly prices compared to similar services.
But then the triggers.
Those can be webhooks. You can find a full list of webhooks on which you can subscribe here: https://shopify.dev/api/admin-graphql/2021-10/enums/webhooksubscriptiontopic For "welcome", there is no actual webhook. You have to track this yourself. When your callback URL hits you can check if you have an instance of that store in your database, if not, you know they're new and you can send them a welcome email.
For "billing", there also isn't a webhook I believe. But when a merchant accepts a charge, they're redirected to a URL you have passed with the request for a billing URL from Shopify. When the user hits that, you know they've accepted the charge.
And I don't know what you mean by "unsubscribe" but if you mean "uninstall" then there's a webhook for that.