Is the system $y[n]=x[n]+2=T\{x[n]\}$ an LTI-System? Is the system $y[n]=x[n]+2=T\{x[n]\}$ an LTI-System? 

Linearity:
$ \alpha \cdot T\{x_1[n] \} + \beta \cdot T\{x_2[n] \} = T\{\alpha \cdot x_1[n]\ + \beta \cdot x_2[n] \} \\\alpha \cdot (x_1[n]+2) + \beta  \cdot(x_2[n]+2)= \alpha \cdot (x_1[n]+2) + \beta  \cdot(x_2[n]+2)$ 

Time-Invariance:
$y_1[n] = y[n-n_0] = x[n-n_0]+2 \\ x_2[n]= x_1[n-n_0]\\ y_2 [n]=T\{x_2[n ]\}= x[n-n_0]+2 \\y_1[n]=y_2[n]$ 

So I would say it is an LTI-System, is that right?
A system that adds a constant is rarely linear. If you have two inputs $x_1$ and $x_2$, each one gets the constant, so their combination get the constant two times. While the input $x_1+x_2$ gets the constant only once.