Very basic questions about linear systems
The state-space model of a causal continuous-time LTI system is

$$\begin{align}\dot{x}(t)&amp;=Ax(t)+bu(t)\\y(t)&amp;=cx(t)+du(t)\tag{1}\end{align}$$

where $x(t)$ is the state (vector), $u(t)$ is the input, and $y(t)$ is the output.
If the state $x(t)$ is a length $N$ vector, then $A$ is an $N\times N$ matrix, and $c$ is a $1\times N$ matrix (or row-vector, if you like).

In your example, $b=0$, $c=0$, and $d=0$. So there is basically no input and no output. We can only observe the state, i.e., the "internal behavior" of the system. We can only talk about linearity and time-invariance concerning the states. If $A$ is constant, the system is linear and time-invariant in that sense. But since we can't observe a response to an input signal it is impossible to talk about linearity and time-invariance of the system in terms of its input/output behavior. And, consequently, the corresponding tests become pointless.

And concerning point 4. in your question: yes, an LTI system must have zero output for zero input. It is a non-zero initial state that can produce a non-zero output for zero-input, but with a non-zero initial state, a system is not LTI anymore. If it is LTI otherwise (apart from the non-zero initial state), then it can be modeled as LTI by representing the non-zero initial state as an additional input to the LTI system (which must now have a zero initial state, otherwise it couldn't be LTI).