You can get babies by lots of IR connecting - not just the Matchmaker bringing a match for your tama.
If you connect a lot with another tama of the opposite gender they will become best friends, then when they are both Adults (within 24hrs) if you connect again they will have babies - one for your tama, one for the other tama you've connected with. Lots of ppl get their next generation of tamas that way
Tamas will normally turn into "Old Timers" - Otokitchi = female / Ojitchi = male if your tama has not found a mate by connecting or says "No" to the LOVE? question from the Matchmaker. (The Matchmaker is in fact the Otokitchi).
I am not
totally sure, but the Matchmaker starts visiting the tama three times a day when your Adult has been an adult for 72hrs without finding a mate. If you keep saying No to the Matchmaker she stops visiting - after 2 or three days (so you have about 6-9 chances to say Yes to Love).
After she stops visiting, your tama will turn into an old character within a few hours.
Once that happens, the only tama you can mate with is another old timer of the opposite gender by connecting. You will not get a matchmaker visiting Old Timer tamas.
Phew... that was a bit of a long, rambling reply... hope it makes sense..