Is Selichot said in the house of an avel during shiva on the 10th of Tevet? Does the mourner lead the Selichos?
Nitei Gavriel Aveilus1 95:14 brings as follows.
Turei Zahav Yore Deah 384:1, Derisha, and Pri Megadim say that Selichos is recited in the home of a Aveil. Shaalos U'Teshuvos Yehuda Yaaleh Yore Deah 353 says that the Aveil should also say the Selichos.
Aveil Hashitim page 45 - 6, Machtzis Hashekel 131:10, say that although the Aveil says Selichos, the Aveil should not say the Vidui.
On the next page he mentions that Gesher Hachaim page 204 -4 says to do an abridged version, and that the Pri Megadim says in one place that the Aveil himself should not say and in another place that the Aveil himself should say.
Based on this I would think that the Aveil would not lead the Selichos, unless the Aveil is saying the complete Selichos.
According to what is stated in More than a Tear by Yig'al Sigal (2001),
Selichot are abridged. Everything is said, except for omitting the section Zechor Rachemecha which appears before Shema Koleinu. Shema Koleinu is said, and the rest, until the end is omitted.
This applies on all fast days where Selichot are said. Tish'a B'Av (Some nuscha'ot do say Slichot on Tish'a B'av. See beureihatefilah.com) and the Slichot said around the High Holiday period may have different rules.