Today I have accidentially discovered something that helped my Xperia a little bit.
I started to look into its lag and performance problems after playing for a while with my wife's Motorola Moto G.
By the way, Moto G is a small miracle I'd say.
So I started to investigate for potential source of problems for Xperia. Then... I disabled the "Google search" app (yes, that one on the top of main screen). And the phone works much, much better now !
This is no problem for me as I never used this search bar. But just to confirm, maybe others could also try out if it helps them, too ? It's easily reversible.