Sam,
1. There is a similar machloket regarding shaking hands with a woman. Both Rav Shimon Schwaband his brother asked shailot (of different rabbanim). One was told yehareg vealyaavor and the other was told muttar. Regarding an individual this is fine and common but the Army obviously needs a unified pesak. The address is obviously the IDF Chief Rabbi, who is the mara d’atra (so paskened Rav Yaakov Ariel).
2.From the Chief of Staff’s tone (I heard him on the radio) and the fact that he created a stir by making a joke about it I gathered that he does not want this but needs to do it because of pressure from feminists and singers who for some reason feel personally offended. Thus, even if it is assur it is hanaat atzmo.
Regarding giving a heter, many times issurei derabbana are allowed mishum aiva. Here it is even easier as there are heterim which already exist. One is the microphone. The Seridei Esh mentions zemirot and shirei Eretz Yisrael, which might apply to the type of songs sung at military ceremonies (and the purpose of these ceremonies is morale, which is very important to combat soldiers). Another, put forth by Rav Shlomo Aviner, who is not known as a meikal on tzniut, is that any hanaah the soldier receives is against his will and therefore not halachically hanaah. There is a similar sheetta which says that any melacha one is compelled to do on Shabbat is a melacha she’ain tzericha lagufa and therefore according to Rabbi Shimon (according to whom most Rishonim pasken) it is an issur derabbanan. The Chatam Sofer, the Minchat Yitzchak and others use this to allow giving medical treatment to a Goy on Shabbat mishum aiva (Rav Gustman added that the doctor would lose his license and thus be unable to treat Jews). Rav Yosef Carmel of Eretz Chemda posited that walking out would be embarassing the singer in public. Rav Metzger, who himself does not walk out when women sing at ceremonies, said that he takes his mind off her by saying Tehillim to himself (although he added that this is a difficult heter to give to a bachur).
3. The Gemara is talking about an eshet ish. The doctors started with asking to allow him to be with her. This is clearly not allowed even for pikuach nefesh. Later they tried to bargain and went down to a conversation and just seeing her in the altogether. Obviously, this was just an excuse.