The matchmaker will come as soon as you tama has been a adult for 72h. You do not need another tamagotchi for the matchmaker to come. And, soon as they have breeded, the tama that the match maker found for you will dissapear and you will never see it again.
Then, once breeded, you will have your adult tama and your tamababie on the screen untill 2 days after breeding, and then the adult will leave and you will start your next generation!