The matchmaker visiting times are 10:30 AM, 3:00 PM, and 7:00 PM. She will visit at these times for three days, begining a couple of days after your Tamagotchi becomes an adult.
What do you mean by "can you turn in the matchmakers character"? If you mean your Tamagotchi turn into the same character as the matchmaker, then the answer is Yes, if your Tamagotchi character is female. Just reject all offered mates, and a couple of days after the last time the matchmaker visits, your Tamagotchi will turn into an oldie, like the matchmaker.