On the stability and causality of a discrete system On  MIT's open course a simple exercise with two questions is given. On the first part, they question about the properties of the following discrete system:

$$
y[n]=x[n]+0.5y[n−1]−2y[n−2]
$$

The answer states that none of the following properties can be determined, as no initial conditions were given: stability, causality, linearity, time (in)variance.

I would be inclined to say that the system is at LTI, as the coefficients are constant and the output is simply defined by a recursive relationship to itself and the input. Both poles are inside the unit circle and the system should be stable if it is causal, as the ROC includes the unit circle.


My first question is therefore: could you come with a(n) example(s) of initial conditions (or any argument for that matter) so that the system is: variant in time OR unstable OR not causal OR non-linear? The only conditions that I can think of that could make it behave strangely would be extreme ones such as +/- infinity. I assumed this was not the intent of the initial question. 


Next, the following transfer function is given (unrelated to question above):

$$
H(e^{j\omega }) = \frac{1 - 2e^{-j\omega }}{(1 - \frac{3}{4}e^{-j\omega })(1 - 3e^{-j\omega })}
$$

Here it is also asked about its properties. On the final answer they state that, given that a transfer function is given, one can imply that the system is linear and time invariant. So far so good. Next, however, it states that the system is both stable and not causal, with the following argument:

...Further, from the existence of a frequency response you may conclude that the frequency response converges on the unit circle, i.e., that the system is stable. Given that the system is stable the region of convergence is also implied resulting in a two-sided sequence, meaning that the impulse response is not causal. 

Hence my second question:


How was this concluded from this transfer function, given that no ROC constraint is there?


Thanks in advance.
To your first question, a linear system is causal if we have initial conditions of rest that is zero. This is a sufficient and necceassy condition. This property doesn't  require time invariance

Stability is a consequence of absolute summability of system response. It doesn't depend on the initial condition (of course have to be finite to make any sense).

To your second question the transfer function has a pole at z=3 and it is given that the system is stable. That means the ROC has to include the unit circle. That means that the ROC is to the left of the pole at z = 3, since ROC is towards the left of outermost pole hence system is not causal