For example, can one learn Torah on the fast, since it's really 10 Av? Are any of the usual ways of mourning omitted? Are we more lenient in general in this year's case?
The only difference mentioned in Rishonim that I'm aware of regards a circumcision on Sunday 10 Av. The Shulchan Arukh (OC 559:9) rules like the Mordechai that the members of a circumcision (father/self, Mohel, Sandek) can break their fast in the afternoon after Mincha. Most Rishonim though don't mention this and indeed many communities do not practice it (Arukh haShulchan ibid., Peulat Tzaddik 3:147, Teshuvot veHanhagot 2:252). In many places the circumcision is done at the very end of the day with the meal following nightfall, just like any other Tisha Bav.
Some recent authorities have extended this leniency to various cases of illness or weakness to varying degrees, but this appears to have been a minority view. See discussion here.
The Magen Avraham (OC 557) quotes the Shelah that Nachem is said at all prayers when fasting on 10 Av, since one of the reasons why some only say it at Mincha on regular Tisha Bav is that the Temple only started burning in the afternoon of 9 Av and continued through 10 Av. This is quoted by a number of later sources, though the Levushei Serad claims that this only applies when fasting on 10 Av in addition to 9 Av, so it may not apply in your case.
For the general population though all the regular restrictions (eat, drink, shoes, Torah study, etc.) apply as usual. Tisha Bav is not a Torah command, and the same Sages who instituted it on 9 Av in general instituted to observe it on 10 Av in this sort of year.