BTW, In OC 109 I believe, Reb Moshe says that one who disagrees with him is not only allowed to pasken differently, but is MECHUYAV to do so.
Helpful, as people have stated here umpteen times, the halacha is that only derech chiba is assur. Reb Moshe may feel that a handshake is derech chiba, yet many other fine Rabbanim may feel that it is not. It really is very simple.
As far as the teshuvah about being friends, again it is complex. I believe Reb Moshe found an Avos Dreb Nassan which says Yachol yechabkenah vyinashkena vyidaber imah devarim betailim, T”L Lo sikrevu. Reb Moshe wants to learn that yidaber devarim btailim is inclided in the deoraisa of Lo sikrevu. Yet in another teshuvah, he goes into a very long arichus about yachol yisaper imah meachorei hageder. There he brings a kashya from one of the acharonim who asks where did we ever find an issur on dibur? The acharon remains with a kashya. Furthermore, he brings from a Maggid Mishna who discusses the Rambam that mentions that gesturing, kalus rosh, and many other things get makas mardus. The Maggid Mishna says it is Makas Mardus because it they are all drabanan. So it is not clear that speaking is deoraisa. It would seem that the devarim betailim of Avos Drebbe Nasan is referring only to intimate talk about things that relate to tashmish. This is rather clear from the other Reb Moshe about meachorei hageder, where he explains that it is because it is actually the preparation for tashmish that the fellow wanted. That is why chazal forbid it.
So it is not clear that mere friendly conversation with someone of the opposite gender is forbidden.
Furthermore, as I asked earlier, what if the two of them wanted to get married when they were older? It would then be considered dating which is ok. I know of other cases where people married next-door neighbors who grew up together. One is the daughter of one of the well known Chareidi maggidei shiur on all the national Dial-a-shiur networks who married the learning son of another Chareidi maggid shiur in a Yeshiva. It happens, and is actually a cause for much simcha.
In the case of Reb Reuvain, the story appears in the Reb Moshe Artscroll biography. Her father suddenly passed away very young, and she would often be in Reb Moshe’s house. He was close with the family, and did the tahara for her father.
Helpful, you need to acquire a taste for the depth and complexity of halacha, and the constant counterbalancing factors that go into one of Reb Moshe’s teshuvos. He could either say the Avos dReb Nassan was an asmachta to make it agree with the Maggid Mishna, but there was a Shach that he wanted to make the Maggid Mishna agree with, so he learned the Maggid Mishna that the makas mardus was not because of being derabanan, but because of lav sh’ein bo maysa. Clearly this is not pashut pshat in the Maggid Mishna. Reb Moshes gadlus was in trying to weave a path through his tremendous bekius which often brought him into contradictory situations which he had to iron out.
Those who think a Reb Moshe teshuva is only his last paragraph totally miss his gadlus. Today, people are brainwashed not to think, but to read Reb Moshe’s conclusion like the pope is speaking and scream Daas Torah, end of story. That is not the way Reb Moshe himself wanted them to be learned. He states he has a lot of hanaah when people actually take the time to read the teshuva and debate with him. The gadlus is in his methods.