'About' (BE, vs. 'around' AE) isn't wrong to my ears, but it's also not necessary.
Verbs like 'loll' or 'laze' are possible, but neither of those is a very everyday word to my ears. Both mean simply to relax, to take a rest, to be in a relaxed posture. They say nothing about the physical position of the body; it could be either sitting, or lying on one's stomach, back, or side.
'Loaf' is similar, but not related to posture at all -- you could actually loaf standing up and walking around town. It simply means doing nothing useful, being lazy, wasting time. 'Lounge' is similar to that, but usually at home, in a relaxed setting. To me those don't seem as close; not wrong, but not exactly what the German verb (if I understand it) seems to mean.
'Slouch' seems wrong to me, at least for the picture in #1. Partly, it does mean a specific posture -- but leaning forward, with hunched shoulders, slumping, not lying back in a relaxed way as in the photo. Also, it doesn't mean relaxed in a positive way, but as a negative judgment, implying laziness as a bad habit: not comfort but sloth. For example, a parent, teacher, or military instructor might say 'Don't slouch! Stand/Sit up straight!'
The verb I thought of when looking at the picture (and which I thought I had mentioned in a previous discussion of fläzen that I can no longer find) is 'sprawl,' which means to be in a relaxed, almost lying posture with one's arms and legs spread out, covering more space than if you were just sitting or lying normally. I would say that the guy in the picture is lying sprawled on the beanbag; you can also sprawl, or be or lie sprawled, on a sofa, chair, or bed.