I went through there 2 years ago. No Carnet needed. But quite a few fees: fuel surcharge, "permit of passage" for $15 (why is there such a thing called "visa", which I've already paid for?). All with proper receipts though!
Getting the Turmen visa was a pain in the backside. Once the visa agencies heard the word "Turkmenistan", I was told that I'm on my own. Basically, they're all fed up with dealing with Turkmenistan embassies. I would soon find out why.
UK/German embassy were extremely unfriendly. Embassy in Vienna was very friendly and helpful on the phone, so I went via them.
In 2008 (according to ALL embassies of Turkmenistan I tried), to get a TRANSIT visa you'd need a visa for the country you'll be coming from, AND the visa of the country you'll be going to. (Just in case it's always been your dream of becoming an illegal immigrant there
Only then were you allowed to apply for the transit visa.
A Letter of Invitation is NOT needed. However, 1 week after I applied at the embassy in Vienna (when I was already travelling & about to pick it up en-route), they told me that they changed the rules and that now I DO need a LOI.
After contacting 10 tour operators in Turkmenistan to get me a LOI, they all said: a LOI for a Transit visa doesn't even exist.
Another week (and dozens of emails) later, the Vienna embassy changed the rules back to "no LOI needed", and I got the visa... An exercise in madness!