Setting user input as value?

Disclaimer: Author has updated the question, so my answer only reference second error. Although first error has same cause.

Your error

Error 2 A field initializer cannot reference the non-static field, method, or property 'NumberGuessingGame.frmMain.inputGuess'

is in this line:

int inputGuess = Convert.ToInt32(txtGuess.Text);

That means that you try to assign to field (variable stored in class) non-constant value(apart from object creation).

To avoid this, assign it in your button click handler.

But there are few more issues:

1. Your user input should not go as constructor parameter, because each guess will lead to creating new object. You need to add it as parameter to your checkGuess.
2. You should not generate random number each time user try to guess.
3. You have counter of lost games, so you need a way to reset your game. Maybe you should add another button for new game, but I will suggest to reset game each time you end it.

Here is version of your game class with my suggestions:

public class GuessingGame
{

    int guessesLeft = 0;
    int gamesPlayed = 0;
    int gamesWon = 0;
    int gamesLost = 0;
    int myGuess = 0;

    Random rand;
    int number = 0;

    public GuessingGame()
    {
       rand = new Random();
       number = rand.Next(1, 100);
    }

    public int Guess
    {
        get
        {
            return myGuess;
        }
        set
        {
            myGuess = value;
        }
    }

    // Reset game method. You can call it from form too.
    public void ResetGame()
    {
        number = rand.Next(1, 100);
        guessesLeft = 0;
    }

    public void checkGuess(int newGuess)
    {
        myGuess = newGuess;

        if (guessesLeft > 10)
        {
            Console.WriteLine("you lose!");
            gamesLost++;
            gamesPlayed++;
            ResetGame();
        }
        else if (myGuess > number)
        {
            Console.WriteLine("your guess is too high, try again.");
            guessesLeft++;
        }
        else if (myGuess < number)
        {
            Console.WriteLine("your guess is too low, try again.");
            guessesLeft++;
        }
        else
        {
            Console.WriteLine("you win!");
            gamesPlayed++;
            gamesWon++;
            ResetGame();
        }
    }
}

And your form will look like this:

public partial class frmMain : Form
{
    public frmMain()
    {
        InitializeComponent();           
    }

    GuessingGame myGuess = new GuessingGame(); 

    private void btnCheck_Click(object sender, EventArgs e)
    {      
        // I suggest to add here textbox text validation
        // so user couldn't input anything apart number.
        int inputGuess = Convert.ToInt32(txtGuess.Text);     
        myGuess.checkGuess(inputGuess);
    }
}

In details about error:

These lines from code you provided in question will be executed in the moment your form are created:

int inputGuess = Convert.ToInt32(txtGuess.Text);
GuessingGame myGuess = new GuessingGame(inputGuess); 

First line is unappropriate(compiler tells us about it with error). But with second line your GuessingGame won't get actual user input. In order to get user input we need to process it after some event which will happen after user write it down in textbox. That's why you should do such things in event handlers methods.