Behaviour of System by Impulse Response
The impulse response characterizes an LTI system completely, in the sense that the response to any input can be computed from the input and from the impulse response  by convolution. That's a consequence of the system being linear and time-invariant.
If we define stability as bounded-input bounded-output (BIBO) stability then stability can be determined from the impulse response. For the system to be stable, the impulse response must be absolutely integrable.
Note that when describing systems by impulse responses, or, equivalently, frequency responses or transfer functions, we focus on the input-output relation. We do not consider the internal dynamical behavior of the system. A state-space model is needed if we not only want to describe the system's input-output behavior but also its internal behavior.