Private Equity describes any trading in non-listed companies. Private Equity funds sometimes do the things you’ve described, but private equity just means the shares aren’t traded publicly, it doesn’t denote any particular investment strategy. There’s pros and cons on each side, but if you want liquidity, then you’re going to struggle getting it through private equity investment.
Yes it is. But liquidity in terms of trading volume, not in terms of money the company has in the bank. Liquidity in terms of how many shares are traded each day. If you have shares in a private company, and you want to sell them, you’re going to have to arrange a private transaction, and rely on somebody elses valuation. Going public is one way of addressing that problem.