Testing for Linearity and Shift-Invariance
Your demonstration of shift-invariance is not correct.
You time-shift the input signal by $n_0$, which is correct. But you also time-shift the sinusoidal, and this only happens if you were time-shifting the output.
The description of the system contains a signal that explicitly depends on time ($\sin(2\pi/7 n \cdots)$). In general this is a red-flag for non time-invariance.