We spent the night next to the crater on November 6th, on our way from Konye-Urgench to Ashgabat.
You don't need to go to Dashoguz from Konye Urgench to Darvaza. Actually, if you're coming from Khiva, I'd recommend going to Nukus the day before to visit the Savitsky museum (see here for Khiva-Nukus: viewtopic.php?f=16&t=10517&p=34738#p34738
), then cross the Hojayli-Konye Urgench border and go directly to Darvaza from there.
Once we crossed the Hojayli-KO border (see our report here: viewtopic.php?f=9&t=826&p=34739#p34739
), we were taken to the taxi meeting point next to to the Dash Mosque Museum on maps.me, where one of the taxi drivers drove us to his friend's house where we could exchange our dollars at the better rate of $1=18 manat. On his way back from his friend's house, the taxi driver dropped us for free at the Turabek Khanum Mausoleum. From there, you may need to pay 21 manats/person at the ticket booth and then you can easily walk to all the historic ruins of the old imperial city. All the interesting sites are marked on maps.me, and the 1-hour leisurely walk took us to the main road where it was super easy to hitchhike for free back to the taxi station, where you can see the last two interesting buildings (Nedjmeddin Kubra Mausoleum and Sultan Ali Mausoleum). I would also recommend, walking north to the bazaar from there (about 10 minutes) where you can buy a cheap lunch. That's where we met the friendliest people in Turkmenistan.
Back to the taxi station, you can pay for a seat in a shared taxi to Ashgabat (150 manats/seat) and get off at Darvaza on the main road (which is about 8km from the crater). We negotiated a private taxi for $40 + a pack of cigarettes to take us to the crater, sleep there and take us to Ashgabat the next morning. The car was super large and confortable and could have easily fit 6 people, but we were only 3 to share the cost. Like thisjd above, we also had the experience of a driver who put the music very very very loud and was not particularly friendly but our driver's name was Oibek.
At the crater, we could have slept in the car but slept in a yurt instead. There are a lot of yurts and they are very close to the crater so you do not need to worry about accommodation there. We slept there on November 6th so they are open late in the season (it was really cold at night already when we stayed there), if not all year long. We slept in a yurt in one the 'more fancy' yurt camp for $10/person (no food included as we bought our one in the excellent Konye Urgench bazaar), but there are also local independent guys renting their own yurt for $5/person. The guy we traveled with did that but he slept on the floor next to the yurt owner.
The crater is really amazing at night and it's a very unique experience I'd recommend to anyone. However, I'd strongly recommend spending the night there, seeing it only in the daylight is like 10% of the experience in my opinion.
The next morning, we left around 8:00 and arrived around 11:00 in Ashgabat. Our taxi driver wanted to drop us at the taxi station, which is inconveniently located 15km north of the city. We insisted that this was not the deal and threatened not to pay him full, so he paid another taxi driver (I think 30 manats) to take us to our hotel.