Hi Andre,
So, there are several issues here, the most important to address to start is the fact that as far as billing goes, you're responsible for anything you deploy on the platform. Having a manual scaling module always up and making requests constantly will definitely tend to increase your use of billable resources.
Now to get to the technical details, from reading the docs it's clear that the push queue documentation doesn't recommend a manual scaling module in this manner at all. The use of push queues is best combined with an automatic scaling module, so that the the tasks which are pushed will trigger requests on that module, and it will spin up or down depending on the volume of tasks to process. I think you're thinking of pull queues if you've implemented something to check if queues are available.
As for your last question, you don't need to send push queue tasks to the default module. You can set a module target when you create the task, and this is described in the docs page you linked (ctrl + f "target").
Let me know if I've misunderstood anything about your post or if you have any questions beyond this and I'll be happy to help.
Best wishes,
Nick