How do i know it is will be periodic
The easiest proof uses the convolution property of discrete LTI systems.
for Continuous time:
If you write down your periodic sigal as it's Fourier series (with omega =2pi/period) and describe the output using the convolution property you have:

We get the output as Fourier series with the same period.
now the exact same thing can be applied for discrete system (by using the discrete convolution).

2)What conditions are required for this system to be linear?.

difference equations are analogous to differential equations. With the initial conditions at rest (if x(t)=0 for negative t then y(t)=0 as well) The differential equation describes LTI system.
The same can be shown for the discrete time case.
With initial rest Your equation will describe LTI system.