Hi Spyros,
For 1), you can add a "link" as an activity in the 4 courses that takes learners to the scheduler. If you want something more interactive, you can also create a button with bootstrap: https://getbootstrap.com/docs/4.0/components/buttons/
For 2), I think the easiest way would be to ask students to write which course they're enrolled in the "message" box in the booking form.
There are multiple options when it comes to the grades, I think the simplest one would be to add a grade item in the gradebook. However, teachers will need to add the grades directly in the gradebook/with a .csv file. Do you think that this is feasible for them?