There are two branches as long as I know now, stable one and development one.
Yes, but the previous week's video had 01.144 for stable and 01.145 for development. The title of that video has been changed to 01.140 & 01.144, but the video itself wasn't edited and 01.145 is still being shown in it, as well as in this week's video.
I'm currently in the development branch, however I haven't launched the game for some month now, so I didn't see actual version numbers after the updates.
a PC slightly above the minimum requirements? Is it possible to render planets? If yes, of which size? What's the critical part: RAM, video card...?
The minimum on the store page can't handle planets as also can't handle current versions of the game in empty space (the presented there NVIDIA GeForce 8800GT / ATI Radeon HD 3870 are DirectX 10 capable cards).
For planets 8 GB and a fast swap file is the minimum
(at least when using the default world scenarios - it might be working better if you spawn a single planet manually in creative and then switch to survival to play on it, instead of playing on a full set of planets from the world definition).
Newest processor model isn't that much required. The game will run quite fine on old Core2Quad Q6700 overclocked to 3200 MHz.
I think the graphics card is what matters the most for the game - for NVidia it should be at least a model that ends with 70 (970, 770), or 80 when in older series (580, 680).