Hey BunnyHugger
Not as simple as you might think to answer your questions
Accommodation choices are limited compared to Kruger and what we will recommend depends on what you are willing to spend. I assume you'll not be camping - otherwise you would have mentioned it.
Even though there are way fewer kilometres to cover in the Kgalagadi compared to Kruger I wouldn't just stay in one camp and attempt to do my drives from there unless you don't mind getting in the car for gate opening and only return by gate closing if you want to be able to cover as much as you can. Rule of thumb I'd recommend to split up your about seven nights between at least two camps if you don't want to move too often. If you are okay with the split of 2-2-3 (in any order) I'd even say choose 3 different camps.
If you take a stop over, I'd try to get into Urikaruus or Kieliekrankie for the first two or three nights. If you decide to tackle the 1000 something kays from the Jo'berg area in on go, you might have to stay at Twee Rivieren for the first night - depending on the time of arrival. From the main gate you can reach Urikaruus within 1.5 hours and Kieliekrankie in under an hour if you do the max speed. No time for stopping. If you don't want to do that, you'll stay at Twee.
Why I prefer Urikaruus and Kieliekrankie to Twee? The first two camps are unfenced and only host 10 people each in total. No artificial light at night other than the floodlight at the waterhole and every unit has got a view of that. Urikaruus gets the preference to Kieliekrankie because the waterhole there usually is way busier with all sorts of animals big or small and common or special. The downside is: both camps are hard to come by in terms of bookings - at ANY time of the year...
For the next nights, I'd recommend Nossob. The landscape differs considerably on that side of the riverbed compared to the Auob. Main attraction: lions - close up and personal
Lastly, I'd spend a couple of nights at Kalahari Tented Camp or Mata Mata, with the preference of the first mentioned one. It's another unfenced wilderness camp and is located 4 km south of Mata Mata. Even though there are 15 tents, some of which host up to four people I still prefer it to Mata Mata, especially at gate opening time because you don't get the queues like at Mata. Unless the lions occupy the reception area and you don't manage to retrieve your permit that is...
But you can also go back to Urikaruus or Kieliekrankie from Nossob - or even Twee Rivieren if you want a head start back home.
Thing is, you'll never know where the action will take place. We had years when staying at Twee Rivieren only would have been the thing to do. In other years Nossob would have been brilliant. Then there were times when the area between Mata Mata and Urikaruus were too hot to handle and last time Urikaruus to Kieliekrankie was the place to go.
Mid May generally sounds perfect to me, even if I have never been there myself. I'd go any given time if I could, but I'm restricted to my school holidays over here. Yet, I know quite a few people who love May and the bookings confirm that. Talking of bookings: It is almost impossible to take a spontaneous trip to the Kgalagadi. You will always have to battle at the first of the month eleven month ahead of your planned stay. And more often than not you'll have to be flexible in what you accept as accommodation... (Snap, okie!)
Other than that, I cannot recommend going because it's very addictive and harder to give up than smoking.
So I have a bit more time to plan my KTP trip. (Bit of a bittersweet one that). 
