How to declare a variable such that it exists once per class, kind of like static, but derived classes would also receive a new instances

The simplest solution is to just hide the public static field inherited from the base class:

using System;

public class MyBaseClass
{
    public static int i = 3;  
}

public class MyDerivedClass : MyBaseClass
{
    new public static int i = 1;
}

public class Test
{
  static void Main()
  {
     Console.WriteLine(MyBaseClass.i); // 3
     Console.WriteLine(MyDerivedClass.i); // 1
     Console.ReadKey();
  }
}

I think the best you can do is to keep track of your values in a static dictionary which maps from types to values and then use that table in a public property.  The Set would just retrieve the type of the current object and use it to map into the dictionary and retrieve the value.  The Get would use the current object's type to see if the key already exists in the dictionary.  If not, return the default value, if so, return the value in the dictionary.  Example code:

public class MyBaseClass
{
   protected static Dictionary<Type, int> _mapTypeToVal = new Dictionary<Type, int>();
   public int i
   {
      get
         {
            if (!_mapTypeToVal.Keys.Contains(GetType()))
            {
               return 0;
            }
            return _mapTypeToVal[GetType()];
         }
      set { _mapTypeToVal[GetType()] = value; }
   }
}
 
public class MyDerivedClass : MyBaseClass
{
}
 
class Program
{
   static void Main(string[] args)
   {
      MyBaseClass obj1 = new MyBaseClass();
      MyDerivedClass obj2 = new MyDerivedClass();
      MyDerivedClass obj3 = new MyDerivedClass();
      obj2.i = 57;
      Console.WriteLine("obj1.i = {0}, obj2.i = {1}, obj3.i = {2}", obj1.i, obj2.i, obj3.i);
      obj1.i = 3;
      Console.WriteLine("obj1.i = {0}, obj2.i = {1}, obj3.i = {2}", obj1.i, obj2.i, obj3.i);
      obj3.i = 20;
      Console.WriteLine("obj1.i = {0}, obj2.i = {1}, obj3.i = {2}", obj1.i, obj2.i, obj3.i);
      Console.ReadKey();
   }
}