Billable hours not directly connected to the invoice

Why are the billable hours not selectable in my invoice?
When this is a build in option please explain how I can use this.
It will be nice to have an option to tick off invoices have been payed in the “invoice module” or the “dashboard”

What exactly do you mean? Every line on an invoice can be edited. How do you expect them to be selectable? Have you read the Guide?

Sales invoice status is shown automatically in the Sales Invoices tab. For example:

You don’t “tick off” invoices. Manager displays status when you have recorded receipts against them.