Ayame begins to make her move since she believes Kairi and Momo's relationship is doomed because they have been together for 10 years and are not married. Everyone is taken aback when Momo reveals that she and Kairi haven't even discussed marriage. However, this revelation does give Ayame hope since there's a chance that they might be able to work things out.
Later on, when Kairi goes to visit Momo in the hospital after she was injured in a car accident, they do end up getting married. But first, Kairi takes care of some business with Ayame. When Momo asks why she needs to see her, Kairi tells her it's about their daughter Haru. Since Momo is unable to have children, Haru will be her sister-in-law. At first, Momo refuses but later changes her mind when she realizes how important it is to Kairi that she becomes her wife.
So yes, Momo and Kairi do get married. But don't expect them to live happily ever after since they're still two people in love who can never really be together.
Although they are happy in their relationship, it is not without problems. He marries Levana and crowns her as Commonwealth Empress. Kai returns to the Commonwealth while maintaining his romance with Cinder. Eventually, he realizes that she is evil and leaves her. After this incident, Levana divorces Kai and becomes Princess Ruby of Beacon.
She meets Prince Eric of Atlantis who saves her from drowning and they fall in love. When King Arthur learns that she is still alive, he seeks out Prince Eric to negotiate a peace deal with him. However, when Ruby refuses King Arthur's offer, he declares war on Atlantis. During the battle, Ruby is killed by Merlin after he uses his magic to transform her into an immortal phoenix bird to save Atlantean culture. With her death, the peace between Earth and Atlantis breaks down forcing both worlds to go to war again.
After many years, Kai realizes that he loves Cinder again and they reunite. However, she does not feel the same way and wants nothing to do with him. Desperate, he goes to see Morgana who gives him an idea on how to win Cinder back. Using his new found power, Kai resurrects everyone that he has ever loved and makes them stand in front of him. Then, he creates a world where everything is perfect and they can be together forever.
Kyo expresses his desire for them to be together indefinitely. Tohru and Kyo are shown to have married at the end of the series.
But, to answer your question, MEIKO and KAITO do not have an official relationship and will never have one. They are just friends who go out together sometimes.
Kuroba and Aoko Nakamori are not yet a couple in Magic Kaito Kaito because the manga is not yet finished, but in the final chapter of Yaiba, a scene set after a three-year time skip has Keiko Momoi telling Sayaka Mine that Kuroba and Nakamori are now a couple, and Keiko's friend confirms having seen them together. This means that Kuroba did marry Aoko.
However, this does not mean that they will end up getting married. In Japanese culture, it is common for people to get married before they reach age 20 or even younger if their family wants them to get married so they often run away from home to avoid getting married. Also, in Japan, there is no law that forces someone to get married so they can divorce themselves from their spouse if they want to.
In the English version of the manga, this scene was changed slightly so that instead of saying that Kuroba and Aoko are now a couple, Sayaka says "I see". This change was probably made so that readers of the English language would not think that Kuroba and Aoko were now married since she could have been talking about anything else other than their marriage.
So basically, yes, Kuroba does marry Aoko but they don't stay married.