The Rema (OC 553:2) writes that when Tisha B’av is on (or pushed onto) motzaei Shabbos, one must wait until after saying barchu before changing into one’s non-leather shoes. One should undo them before maariv, and then slip them off, ideally without handling them (Mishna Berura 553:6). Only the chazzan should change them before maariv. R' Shlomo Zalman Auerbach (Halichos Shlomo, Tisha B'Av 15:n15) wrote that when one goes to a Shul that is davening after nacht, they should change their shoes before maariv.
R’ Ephraim Greenblatt (Rivevos Ephraim 1:380:2) writes that one shouldn’t come to shul on Shabbos wearing non-leather shoes because one isn’t allowed to display mourning on Shabbos. Likewise, one can’t carry them on Shabbos, as that would be hachana, preparing for after Shabbos. Thus, one should bring them to shul before Shabbos. (ibid 3:356; 3:606:7).The berachos on havdala are split up. One doesn’t say hagefen until Sunday night when one can drink wine as it is no longer the nine days (See Mishna Berura 556:3). One doesn’t smell the besamim or say the pesukim at the beginning. While most people say the beracha over the candle in shul before eicha, R’ Shlomo Zalman Auerbach (Halichos Shlomo, Tisha B'Av 15:n22) suggests that husbands wait until they get home and say it for their wives, too. One who isn’t fasting should ensure that they say havdala (with drinking wine / chamar medinah) before eating (Shemiras Shabbos Kehilchasa 62:46).