C# – Exact use of Abstract class

string is an alias in C# for System.String.
So technically, there is no difference. It's like int vs. System.Int32.

As far as guidelines, it's generally recommended to use string any time you're referring to an object.

e.g.

string place = "world";

Likewise, I think it's generally recommended to use String if you need to refer specifically to the class.

e.g.

string greet = String.Format("Hello {0}!", place);

This is the style that Microsoft tends to use in their examples.

It appears that the guidance in this area may have changed, as StyleCop now enforces the use of the C# specific aliases.

Yes, an abstract class can have a constructor. Consider this:

abstract class Product { 
    int multiplyBy;
    public Product( int multiplyBy ) {
        this.multiplyBy = multiplyBy;
    }

    public int mutiply(int val) {
       return multiplyBy * val;
    }
}

class TimesTwo extends Product {
    public TimesTwo() {
        super(2);
    }
}

class TimesWhat extends Product {
    public TimesWhat(int what) {
        super(what);
    }
}

The superclass Product is abstract and has a constructor. The concrete class TimesTwo has a constructor that just hardcodes the value 2. The concrete class TimesWhat has a constructor that allows the caller to specify the value.

Abstract constructors will frequently be used to enforce class constraints or invariants such as the minimum fields required to setup the class.

NOTE: As there is no default (or no-arg) constructor in the parent abstract class, the constructor used in subclass must explicitly call the parent constructor.