If you buy into an IPO you're helping the business raise funds. If you buy stock from other stock owners it doesn't help the business.
Additionally the people who own shares own the company so what benefits the shareholders benefits the company and vice versa as their interests are the same.
Incorrect. That's what keeps the market for the stock liquid, and that contributes to value which the company can exploit through future stock issuing. Not to mention the indirect benefits to stakeholders within the company in terms of price and liquidity.