Property (T) and subgroups of finite index

Yes. For groups $H\subset G$, with H a lattice, H has (T) iff G has (T). When both groups are discrete being a lattice is the same as being finite index.

Almost every thing you ever need to know about Property (T) can be found here

http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf

I think even more is true. See Proposition 2.5.5 in "the book":

Bekka, Bachir; de la Harpe, Pierre; Valette, Alain (2008), Kazhdan's property (T), New Mathematical Monographs, 11, Cambridge University Press, MR2415834, ISBN 978-0-521-88720-5, http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf

for (I think) a stronger result about property (FH)