(1) Yes. This will typically be exercise price * no. of options. You are exercising your rights to convert your options into shares of the company.
(2) Yes, that's mostly it. Current share price is calculated by fair market valuation of the company. For non-listed companies, this would either be generally available in your Carta/Equitylist dashboard or the company will share this number with you at the time of the buyback. You could reach out to your finance/HR/equity team to understand this number before the buyback.
(3) Yes and yes. There's perquisite income tax when you exercise the options and there's capital gain tax on selling the shares in a buyback/public market.
(4) In most cases, there's a buyback before an IPO. The company will list the shares at a higher price than your share price. In case that doesn't happen, you are a public equity holder post the IPO and your share price can increase/decrease as the market moves.