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.
You need to have Cookie Clicker as your active tab to let Raptr track your gametime and the browser doesn't need to have focus. If you're browsing with multiple tabs, everytime you switch out from Cookie Clicker, Raptr will notify you something like 'Cookie Clicker - 9 seconds played'.
This behavior has been tested with Chrome and Firefox. With Explorer I'm under the impression that gametime isn't tracked at all.
Best Answer
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.