My question, at the bottom line, is what is the appropriate(best) way to manage our connection towards MySQL db with C#.

Well, currently I'm working on some C# (winforms type) <-> MySQL application and I've been looking at Server Connections in MySQL Administrator, been witness of execution of my mysql_queries, connection opens an closes, ... an so on! In my C# code I'm working like this and this is an example:

public void InsertInto(string qs_insert)
        {
            try
            {
                conn = new MySqlConnection(cs);
                conn.Open();

                cmd = new MySqlCommand();
                cmd.Connection = conn;
                cmd.CommandText = qs_insert;
                cmd.ExecuteNonQuery();
            }
            catch (MySqlException ex)
            {
                MessageBox.Show(ex.ToString());
            }
            finally
            {
                if (conn != null)
                {
                    conn.Close();
                }
            }
        }

Meaning, every time I want to insert something in db table I call this table and pass insert query string to this method. Connection is established, opened, query executed, connection closed. So, we could conclude that this is the way I manage MySQL connection.

For me and my point of view, currently, this works and its enough for my requirements.

Well, you have Java & Hibernate, C# & Entity Framework and I'm doing this :-/ and it's confusing me. Should I use MySQL with Entity Framework?

What is the best way for collaboration between C# and MySQL? I don't want to worry about is connection that I've opened closed, can that same connection be faster, ...