Aside from buying IPOs, what exactly is the point of shareholders?
The shareholder system is a way of distributing the ownership of the company - each shareholder owns a certain part of the company. Shareholders are the owners of the company, in several company legal forms, such as an LLC. They exist before and after an IPO (Initial public offer), meaning that both public (traded) and private companies may have shareholders.

The shareholders can fire and hire the Management team (CEO/CFO/ etc), which answers your questions 2 and 3. They usually signal the Management team which actions they expect the company to take to maximize the company value, reflected by the share value. If the shareholders have a short-term gain objective, this might harm the long-term sustainability of the company - a case where the management team and the shareholders might have different perspectives.

Raising funds through the sale of shares is attractive for companies because it is interest free, and might give you access to much more funds than you would ever be able to get from bonds - depending on how a company communicates its future potential. However, this comes with a loss of ownership. There is only so much unsecured debt one can sell - and debt is always paid back with interest.