You are really twisting things around to try to make this scenario fit your definition of deus ex machina.
The 'problem' in question is pretty minor with many, many solutions possible. No way was ssamba boxed into a corner when she decided on this one. It might be a bit cliche, but coincidences do happen in real life. This isn't even the first time No-Rae has overheard someone talking about her future wife. Remember the bathroom scene with the catty girls badmouthing her? Seol-a is very popular and subject to lots of gossip. I bet No-Rae has even overheard more conversations about her before they were friends but it just never registered in her mind because she's not the gossipy type.
I second this. Everyone involved goes to the same university, so it makes sense that their paths could cross.
Deus Ex Machina comes from greek plays where they had to clear the stage at the end (there was no curtain) and an easy way to do that would be to lower a god down with a machine to tell everyone what they needed to do next and resolve the play. Such as in Euripides' Orestes where Orestes is about to kill Menelaus' daughter, Menelaus wants to kill Orestes, etc. etc. And Apollo pops out to let them know that they have their mythology wrong and Orestes is supposed to marry the girl he's about to slit the throat of, Menelaus is supposed to forgive him, Elektra is gonna marry Pylades, and Orestes needs to go get purified and whatnot like in the official mythology. Then they all leave the stage, but without Apollo showing up the play wouldn't be able to resolve itself and leave the stage (kind of a lame and bizarre ending, because Apollo is talked about throughout the play, but this is his first official appearance and we wanted to see the real action!).
A very obvious modern equivalent to Euripides' usage would be suddenly waking up just as something terribly awful is about to happen in the story and, "Oh look, it was all just a dream huh, it's over."
I don't think ssamba is going to do anything half as lame as that to us.