Without any upgrades or anything, a Golden Cookie spawns m minutes after the last one disappeared, where m is a random number between 5 and 15. Each of the upgrades Lucky day and Serendipity cut that time in half, and the upgrade Gold hoard makes them spawn pretty much constantly instead instead. Also, if your last cookie gave you a cookie chain, then the next one spawns 3 seconds later. This means that you are not more likely to get a Golden Cookie if you haven't clicked on one in a while.
Without any upgrades, again, a Golden Cookie lasts for 13 seconds before disappearing. The Lucky day and Serendipity upgrades both double that. If your last cookie gave you a cookie chain, then it disappears after 6 seconds instead.
When you get what appears to be a flat number of cookies, it either gives you 10% of your cookies or the equivalent of 20 minutes of cookie production, whichever is lower.
The bonus you get is randomly chosen each time you click the cookie. It normally chooses between Frenzy, which multiplies your production by 7, and Lucky, which gives you cookies as I just described. There is also a small chance of getting a Click frenzy, which multiplies your cookies per click by 777, and an even smaller chance of getting a Cookie chain, which gives you 6 the first time you get it, 66 the second time, etc. until you get something else.
Once the Grandmapocalypse begins, you also have a chance of getting Ruin, which makes you lose half as many cookies as you would gain from a Lucky cookie, Clot, which cuts your production in half, and Elder Frenzy, which multiplies your production by 666 for a few seconds.