I have a v 4.5 and here's how my matchmaker stuff works. (this is probably the same 4 v4's)
If you connect with another tama enough the friendship level will grow to the point where they get married or mate or something, resulting in a baby. If you don't find a mate, the matchmaker comes and gives you a mate. The match maker comes after your tama has been an adult for a while, and there is no way to make it come sooner.
I you refuse the partner the matchmaker gives u, then your tama will turn into an oldie and eventually die. Then a new baby will come but it will start over at first generation. If you accept the partner, then the adult stays with the baby for the rest of the day and the full day afterwards. Then, at midnight, it watches it's baby sleep for 1 minute and then does a little dance and floats up int the air and off the screen. The next day the baby is alone and it carries some of the adults skill points. If the adult had
456 funny points, the baby will have
45 funnty skill points to start with. If the adult had
321 gorgeus points, then the baby will have
32 gorgeus points and etc... (i'm guessing that on music star they have different categories 4 skill points, but i'm just giving an example)