What governs how fast grandmas make cookies

The answer to your question

The base amount of milk is 0%. The current formula for calculating milk is milk = achievements / 25. That means that for every achievement, you get 4% milk.

Are you sure 58 is the number shown in the stats page? I currently have 40 achievements, giving me 160% milk (as expected).

I have just discovered that not all achievements count. The August 27th update states:

due to popular demand, retired 5 achievements (the "reset your game" and "cheat" ones); they can still be unlocked, but do not count toward your total anymore.

If you hover over one of these achievements, it'll say [Shadow] in the tooltip. An example of this is:

Uncanny clicker ("Beat the clicking speed world record."):

I got this through using MouseKeys to sort of gain an advantage :P

This means that even though it's displayed in the stats page, it doesn't actually count when calculating milk %. Thus, even though I have 42 achievements, since one of them is a [Shadow] achievement, the game calculates my milk as 41 / 25 = 164%.

More milk facts!

The text next to milk is just flavour text. When milk is less than 100%, it's simply plain milk. When it's at least 100% but less than 200%, it's chocolate milk. Otherwise, if it's at least 200%, it's raspberry juice.

Milk will only affect Cookies Per Second (CpS) if you have at least one kitten upgrade. The first one, Kitten helpers, is unlocked at 50% milk and adds a 0.05 * milk multiplier. (For instance, at 50% milk, your CpS will be multiplied by 1.025.)

The other upgrades are:

At 100% milk, Kitten workers. This adds an additional multiplier of 0.1 * milk.
At 200% milk, Kitten engineers. This adds an additional multiplier of 0.2 * milk.

Thus, at 200% milk with all Kitten upgrades, your CpS is multiplied by 1.1, then 1.2, then 1.4, for a total multiplier of 1.848.

Note: sometimes, for whatever reason, getting a new achievement while owning at least one Kitten upgrade will not change your CpS. To force the game to recalculate your CpS, go to your address bar and type in javascript:Game.CalculateGains() or javascript:Game.recalculateGains=1.

^{Interestingly, a comment in the code says it should be 1% per achievement and that raspberry juice should be strawberry juice...}

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.