is $F(x,n)=x(-n + 2)$ time invariant? My book with answers to proakis and manolakis book says that $F(x,n)=x(-n + 2)$ is time invariant while my calculation shows that it is time variant. I have repeated them many times and I am mostly convinced that book is wrong. Can somebody please check my calculations?

First we shift output signal:

$$F(x, n - k) = x(-(n-k) + 2) = x(-n+k+2)$$

now lets put $x'(n) = x(n-k)$ and check what happen when we input shifted signal:

$$F(x', n) = x'(-n+2)=x(-n+2-k)=x(-n-k+2)$$

since $F(x, n - k) \neq F(x', n)$ this system is time variant. Is my method correct?
Your reasoning is correct and we arrive at the same conclusion.

The system's output is given by

$$y[n]=x[-n+2]\tag{1}$$

Let

$$x_2[n]=x[n-k]\tag{2}$$

The corresponding output is

$$y_2[n]=x_2[-n+2]=x[-n-k+2]\tag{3}$$

From $(1)$ the shifted output is

$$y[n-k]=x[-(n-k)+2]=x[-n+k+2]\tag{4}$$

Since $(3)$ and $(4)$ are not identical, the system is time-varying.