Can you use reflection to find the name of the currently executing method?

As of .NET 4.5, you can also use [CallerMemberName].

Example: a property setter (to answer part 2):

protected void SetProperty<T>(T value, [CallerMemberName] string property = null)
{
    this.propertyValues[property] = value;
    OnPropertyChanged(property);
}

public string SomeProperty
{
    set { SetProperty(value); }
}

The compiler will supply matching string literals at call sites, so there is basically no performance overhead.

For non-async methods one can use

System.Reflection.MethodBase.GetCurrentMethod().Name;

https://docs.microsoft.com/en-us/dotnet/api/system.reflection.methodbase.getcurrentmethod

Please remember that for async methods it will return "MoveNext".