The Rema's sheetah seems pashut -
Aveilut b'farhesya on Shabbat becomes a problem only or primarily when some are aveilim and when others are not. All bets are off, however, when ALL are equally aveilim!
- The bensching for aveilim is not said on Shabbat, unless all present are aveilim, then it is recited.
- Same with taking off shoes, when aveilim are amongst themselves they may take them off, otherwise it constitutes Aveilut b'farhesya.
Regarding various melodies, Ashkenazi melodies often reflect the season. This is analogous to Shabbat Shuvah and Shabbos hol hamoed which take on their seasonal characteristics. At Breuer's, tunes from forthcoming Yamim Nora'im are introduced as early as Shabbat Nachamu.
For example, singing L'cho Dodi to Eli Tziyyon. It's simply before Shabbat anyway, especially for those who accept Shabbat early during the summer
As far as the p'sukkim in the Haftarah read with Eichah Trop, we do this on Purim too during the reading of the Megillah. It's primarily a manifestation of the p'sukkim's sentiment.