Update many-to-many entity with ASP.NET Core MVC MultiSelectList

I have a form where I want to update an entity with a many-to-many relationship with another entity, and I'm trying to use the select tag to create the MultiSelectList control. I'm using ASP.NET Core 1.0 with Entity Framework.

Say I have the following entities:

public class Student
{
    public int Id { get; set; }
    public string Name { get; set; }
    public List<StudentCourse> Courses { get; set; }
}

public class Course
{
    public int Id { get; set; }
    public string Name { get; set; }
    public List<StudentCourse> Students { get; set; }
}

public class StudentCourse
{
    public int StudentId { get; set; }
    public Student Student { get; set; }

    public int CourseId { get; set; }
    public Course Course { get; set; }
}

The view model:

public class StudentVM
{
    public Student Student { get; set; }
    public IEnumerable<SelectListItem> CourseList { get; set; }
}

The view:

@model StudentVM
<form asp-action="Edit">
    <div class="form-horizontal>
        <div class="form-group">
            <select asp-for="CourseList" asp-items="Model.CourseList" />
        </div>
        <div class="form-group">
            <div class="col-md-offset-2 col-md-10">
                <input type="submit" value="Save" class="btn btn-default" />
            </div>
        </div>
    </div>
</form>

The controller's action methods:

[HttpGet]
public async Task<IActionResult> Edit(Student student)
{
    var viewmodel = new StudentVM {
        Student = student,
        CourseList = new MultiSelectList(
            _db.Courses,
            "Id",
            "Name",
            student.Courses
            .Select(sc => sc.CourseId));
    };
    return View(viewmodel);
}

[HttpPost]
[ValidateAntiForgeryToken]
public async Task<IActionResult> Edit(StudentVM viewmodel)
{
    if (viewmodel.Student == null)
        return NotFound();

    if (ModelState.IsValid)
    {
        Student student = viewmodel.Student;
        var newCourses = viewmodel.CourseList
                         .Where(i => i.Selected)
                         .Select(c => new StudentCourse {
                             StudentId = student.Id,
                             CourseId = Convert.ToInt32(c.Value)
                         });
        student.Courses.RemoveAll(sc => !newCourses.Contains(sc));
        student.Courses.AddRange(
            newCourses.Where(nc => !student.Courses.Contains(nc)));
        _db.Update(student);
        _db.SaveChanges();

        return RedirectToAction("Index");
    }
    return View(viewmodel);
}

Now come the questions...

Question 1

When StudentVM comes into the HttpPost variant of Edit, both the properties are null. Why do the properties in the view model get cleared before entering the Edit function? I've confirmed they are being set correctly in the HttpGet variant, and that the data displays correctly in the view.

Question 2

Since the data coming into Edit is invalid, I can't test my database updates. Is my method of getting the selected items from the SelectList and adding/removing items from Student.Courses correct? I haven't been able to find a lot of documentation regarding many-to-many relationships in EF Core.

I'm rather new to ASP.NET and Entity Framework, so any tips or criticisms would be greatly appreciated.