A1. Who is going to buy them? Someone who thinks that the company will eventually recover and has cash they don't have an immediate need for.
A2. Could a stock become illiquid? Yes. Sometimes the outlook for a company becomes so dire and is so widely perceived that all activity dries up. Exchanges regularly kick off companies that can't support a minimum price or level of activity. Some companies go bankrupt and their stock becomes worthless, i.e. nobody will buy it because the company is no longer a functioning business.
A3. Must companies buy back their shares? Nope. Companies may offer to buy back shares, but they typically they do this when the company is flush with cash and has no better use for the cash. It's an alternative to offering a dividend.
A4. Is there always a buyer? No, not as absolute. But bargain hunters will often speculate on deeply troubled companies, buying stock at very low prices, taking a chance on a (very rare) turnaround, or (more commonly) that the company has enough assets to leaving something behind after debts are covered. There are actually investors who specialize in illiquid stocks.