To qualify as IRS Incentive Stock Options (ISOs), they have to expire shortly after you're no longer an employee; some companies these days offer options than will instead become non-qualified and have a much later expiration, but that wasn't common in the dot-com boom.

For companies that expect to survive for years after the layoff, they may give you X months on payroll, and then Y months severance. Your options expiration because of no longer being an employee would generally start after the X; but I think most of the layoffs then were immediate --- the companies needed you off payroll ASAP, so they could stop paying healthcare and what not.

The contracts can be written with any language. But generally yes, you lose them. If they are vested you can exercise them - meaning you pay your own money to buy them and own actual shares.

If you’ve vested any, which is usually 25% after the first year, and then monthly for the next 3 years. You get to keep those shares if you can buy them at your strike price. If you get laid off, you may only have 90 days to exercise (buy) your vested options before you lose that right. Some companies more recently are extending that time.

In the dot com crash era it was common for your options to have a strike price higher than the stock price (under water) so no sense in exercising them, since they are worth negative.

it depends on the company most companies take away your options 90 days after you leave and they are known as "golden handcuffs" because nobody wants to leave! Some companies like Lyft allowed early hires to keep options for up to 7 years. This can actually be a problem because there are SEC rules limiting private company stock (not option) ownership to 500 shareholders at most. Many companies let you exercise options when you leave by buying them. I did this at one company - but the owners were crooks - and when they raised a new funding round of $3,000,000 they just put it in their personal bank accounts of the CEO + 5 VPs and shut down the company!