AES with linear S-Box

Question

If the AES S-Box is replaced with a linear or affine transformation, for instance the identity mapping $\sigma(x)=x$, does the cipher become entirely affine and hence trivially weak?

Answer 1

does the cipher become entirely affine and hence trivially weak?

Yes; the AES sbox is the only source of nonlinearity (the ShiftRows and MixColumns are linear, and AddRoundKeys for a fixed key is affine), and so if you replace it with a linear/affine one, the entire cipher becomes affine.