(1) In chaotic systems, does the presence of chaos and a strange attractor indicate that there is no equilibrium?
It depends on how you define equilibrium. When starting from a given initial state, the system might display some sort of transient before settling down, so in the vague sense of the system having settled down on some behavior, an attractor would be an equilibrium state - though it's more likely to be called steady state, especially in the case of a strange attractor, which you expect to find in a system with both loss and injection of energy (think of a forced, damped pendulum). If by "equilibrium" you mean a constant state, then obviously chaotic behavior won't qualify.
It's also not uncommon for a system to display multistability, i.e., several final (steady) states, so that it can end up in any of them, depending on its initial state. In such a system, you can have both chaotic attractors and fixed points (but not both on one single orbit of the system, unless you have some other element, such as noise), so, regardless of the definitions one adopts for "equilibrium", the statement "a strange attractor indicate that there is no equilibrium" is in general false.
(1 cont.) I read about steady & unsteady equilibrium but did not understand which gives rise to chaos. Does chaos indicate that the system is not stable?
There are many concepts of stability. If you refer to the stability of a fixed point, then a book lying the table is stable and a pencil standing on its tip is unstable (there's plenty of material about that around, you can start with Wikipedia), in this sense, the link that exists between chaos and unstable orbits is that a chaotic attractor is associated with an infinitude of unstable periodic orbits (search word: horseshoe).
(2) Do chaotic systems become unstable?
The answer to that is given above.
(2 cont.) Is instability + sensitive dependence to initial conditions give rise to chaos?
Unstable periodic orbits and sensitivity to initial conditions are concurrent to chaos. For how it arises I'd refer again to the horseshoe mechanism.
(3) Why does the strange attractor not collapse eventhough it means that the chaotic system is losing energy?
Because it's also being supplied with energy. On a attractor, the system has reached a steady state where both equilibrate.