(a) That's wrong. Being an artificial person with no natural lifespan, a company will continue to exist until it is 'wound up'. Winding up is described as 'voluntary' when, although the company is solvent, the members decide that the company should be wound up and its assets distributed among them. A voluntary winding up can also happen when a company is no longer able to pay its debts (ie it is insolvent) and the members decide that the company should be wound up. To bring about a voluntary winding up, the members seek a court order that the company be wound up.
A 'compulsory' winding up of a company may occur if a court orders that the company be wound up. A court may issue a compulsory winding up order:
when a company is insolvent (unable to pay its debts as they fall due); or
if an application for the winding up is made by an interested person (eg a creditor, or a member, or a director of the company);
and the court decides that it is appropriate that the company be wound up.
When a winding up order is issued, either in a voluntary or compulsory winding up, the court appoints a liquidator, who takes over the assets and control of the company. The liquidator collects all monies owing to the company and sells (realises) its assets. The liquidator then pays the debts owing to the various creditors, to the extent that is possible, and in accordance with the individual creditor's rights. Next, the liquidator distributes any surplus funds to the members (which is unlikely if the company is being wound up because it is insolvent, but possible otherwise). Finally, the liquidator applies to ASIC to deregister the company, thus bringing it to an end.