The first part can be done with the following code:
using System;
class Program
{
static void Main(string[] args)
{
if (args.Length != 2)
{
Console.WriteLine("There must be exactly 2 arguments");
return;
}
int month, year;
bool isValid;
isValid = int.TryParse(args[0], out year);
if (!isValid || year < 2000 || year > 2020)
{
Console.WriteLine("Invalid year");
return;
}
isValid = int.TryParse(args[1], out month);
if (!isValid || month < 1|| month > 12)
{
Console.WriteLine("Invalid month");
return;
}
DateTime dt = new DateTime(year, month, 1);
Console.Clear();
Console.WriteLine("Days of week for {0}\n", dt.ToString("MMMM, yyyy"));

int numDays = DateTime.DaysInMonth(year, month);

for (int i = 1; i <= numDays; i++)
{
if (i > 1) dt = dt.AddDays(1);
Console.WriteLine("{0,2} {1}", i, dt.DayOfWeek);
}

Console.ReadLine();
}
}

As you'll see, the program accepts its arguments from the command line. So, if the program were called dnp.exe, you'd enter at the command line: dnp 2008 8 to see the results for August this year.
The second part can be done with code such as this:
DateTime dt1 = DateTime.Parse("August 1 2008");
DateTime dt2 = DateTime.Parse("August 2 2008");
TimeSpan ts = dt2 - dt1;
Console.WriteLine("The number of days between {0} and {1} is {2}", dt1.ToShortDateString(), dt2.ToShortDateString(), ts.Days);
Yes, the relevant DateTime struct methods/properties do take leap years into account.

Incidentally, the second bit of code works out the span between 2 dates i.e. 1 in the example. To get the total number of days, inclusive, between two dates just add 1.