What's the earliest time (as in exact timing of how long it takes) a Zerg player can have mutas, and the build order to pull this stunt?
Any help appreciated!
Here's a very detailed build whipped up that gets 5 Mutalisks @ 6:36. It's possible there is a slightly faster build for 5 muta, but this will do. As said by other users, I would not recommend this build as it leaves you open to attack for over 6 minutes. If your opponents scouts you once, you will probably die... Might be viable on certain team maps though.
EDIT: I removed the previous 4 Muta build I had because, even though it's faster, anything less than 5 Mutas is worthless. 5 Mutas is the magic number needed to one shot your opponent's workers. If you only have 3 or 4 Mutas you will not be able to adequately harass your opponent, so it's better to wait for 5.
In general, you should always get metabolic boost(speed zerglings) before going Lair tech. That said, here is a build (popularized by idra) that I would recommend using:
This build sets you up with a good economy and allows for any necessary defense or early pressure.
So there is an 8 Minute Mutalisk strategy which does leave you sort of capable of defending yourself until your boys are in the air. There are rumors of a 6:30 Mutalisk strategy, but you're cutting everything from defense to drones.
Both of these builds are pretty bad; and I think its important to discuss why they are so very bad and why you should never do them.
Mutalisks are a fairly high tech unit. In addition to requiring heavy gas, they also require a Lair and a Spire, both of which are Gas heavy and have long build times (not to mention the Queen problem). Even if your opponent doesn't harass you in that time, the normal timing pushes tend to be around 6 minutes. This means that even if you do get your first couple Mutalisks in the air, chances are he's got a strong enough force to just roll you.
Strong Mutalisk builds (ie. not these) usually revolve around having 2 Bases. However, for earlier Mutalisks (6:30) you need to skip that second Hatchery. This means you won't have the production possible to bring about a strong Muta ball, or the resources possible to expand to get to that point.
Finally, you don't see Protoss looking for "Fastest Colossus Possible" builds, because a Colossus requires good Gateway support. Likewise, Mutalisk need strong ground support. The Mutalisk itself is design for hit and run tactics. If your opponent pushes out with a strong MMM or Gateway units, you're going to collapse, and your Muta won't be cost effective.
I realize my trying to dissuade you may seem a little odd since you didn't ask that, but rather how. However, I feel like this question is similar to asking:
How can I stick a Fork in a light socket. If you know enough that it isn't dangerous for you, then you know enough to figure it out on your own.
There are two main limiting factors for Mutalisks:
I did some experimenting and the fastest rush to Mutalisks you get out of this is the following:
Drone x3 9 Extractor (3 Drones mining gas as soon as finished) Drone 9 Spawning Pool Drone x2 10 Lair 10 Overlord Drone x3 13 Spire Drone 13 Extractor (3 Drones mining gas as soon as finished) Drone x3 15 Overlord Mutalisk x3
Here the first three Mutalisks hatch at around 6:07 (Probably even a few seconds faster with perfect execution).
The build order fits together quite nicely and from the point where you start your Spawning Pool there aren't any delays where you have to wait for resources. So with this Spawning Pool timing this is the fastest way possible to get Mutas, and you even get three of them. You are also in a position to continue producing them as fast as gas comes in from your two geysers while getting more drones.
For purely scientific purposes you could tech even faster by starting your Spawning Pool one drone earlier. But to get the tech buildings up fast you'd need to cut a lot of Drones on the way, ending up with no economy and only a single Mutalisk that comes out something like five seconds earlier than the three in the other variant.