SQL server database access helper class
Everything looks good over here, however, I would like to mention some points which cover some of your questions.

I can see your methods are dependent on the global variable connectionString.  


Global variables creates confusion when code grows.
It harder to maintain.
The code which uses global variables gets tightly coupled.


I suggest you to create a class to handle all database connections. This will make your code more flexible: 


It allows to use same connection in other classes. 
It allows to use different database connections.

public class ConnectionManager
{
    public static SqlConnection GetSqlConnection()
    {
        string connectionString = ConfigurationManager.ConnectionStrings["DefaultConnectionString"].ConnectionString;
        var connection = new SqlConnection(connectionString);
        connection.Open();
        return connection;
    }

    public static OleDbConnection GetOleDbConnection()
    {
        string connectionString = ConfigurationManager.ConnectionStrings["OleDbConnectionString"].ConnectionString;
        var connection = new OleDbConnection(connectionString);
        connection.Open();
        return connection;
    }
}

Above method returns open connection so you don't have to open it.
It force to implement using, try-catch-finally or IDisposable to make sure you always close connection which is a good thing.
No need to have constructor for each class which use database.





It is not a good practise to ignore return value. Your method ExecuteNonQuery has return type void which ignores the return value of command.ExecuteNonQuery();
You should use {} to make code more readable.


Before:

public void ExecuteNonQuery(string commandText, CommandType commandType, params SqlParameter[] commandParameters)
{
    using (var connection = new SqlConnection(connectionString))
    using (var command = new SqlCommand(commandText, connection))
    {
        command.CommandType = commandType;
        command.Parameters.AddRange(commandParameters);
        connection.Open();
        command.ExecuteNonQuery();
    }
}


After:

public int ExecuteNonQuery(string commandText, CommandType commandType, params SqlParameter[] commandParameters)
{
    int affectedRows = 0;
    using (var connection = ConnectionManager.GetSqlConnection())
    {
        using (var command = new SqlCommand(commandText, connection))
        {
            command.CommandType = commandType;
            command.Parameters.AddRange(commandParameters);
            affectedRows = command.ExecuteNonQuery();
        }
    }
    return affectedRows;
}


Hope this helps.