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 debug upgrade Gold hoard makes them spawn pretty much constantly 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 15% of your banked 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. As Michael Madsen's answer points out, the effects appear to alternate because every time you click a golden cookie, there's an 80% chance that it will remove the previous effect from the list of possible effects before it picks which one actually happens.
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.
No. The amount of cookies gained from popping a wrinkler is not affected by the multipliers (or, for that matter, the clotting), or the "Click Frenzy (x777)" ones that can also appear.
The number of cookies leeched by the wrinkler (and thus returned when popped) is, however, affected by these multipliers.
Best Answer
Believe it or not, those are still golden cookies.
Their graphic is replaced during the Grandmapocalypse - From the purchase of the One Mind upgrade onwards, you have incurred the wrath of the elders. You can subdue their wrath temporarily with an Elder Pledge and eventually form the Elder Covenant to stop their wrath for the long term. When their wrath is at bay, the golden cookies will return to their previous appearance and behaviour.
While the elders' wrath is incurred, golden cookies have their image replaced with the red one, and two negative effects + one positive effect are added to the possible effects of the golden cookies. This change happens more as you further incur the elder wrath by purchasing certain grandmapocalypse upgrades, starting off changing golden cookies 1/3 of the time with One Mind, 2/3 of the time with Communal Brainsweep and finally 100% of the time with Elder Pact. While there is a chance of negative effects, it's at its worst a little over 50% so the red golden cookies are not entirely unlucky. For more on the percentages, this question's answers cover this in detail.
As I said, to return the golden cookies to their previous behaviour, you will need to: