Checking linearity/time-invariance etc. for recursive system
You made no mistake. Following the hint in Dilip Sarwate's comment, you will be able to show the following input-output relation of the system:

$$y[n]=x[n-1]-x[n-2]+x[n-3]-+\ldots=\sum_{k=-\infty}^{n-1}(-1)^{n-k-1}x[k]\tag{1}$$

Using this relation, showing linearity will be much easier.

Furthermore, note that any system described by a linear difference equation with constant coefficients is linear and time-invariant (if the initial conditions are zero!). And your system obviously belongs to this class.