More playing here on a new machine too and I have it running now.
First I tried the system python2.7 packages again, this went again exactly like yesterday and like @jensens had, too.
You can create a site and five clicks later it dies, checking /var/log:
kernel: [ 1534.593855] python2.7: segfault at 7f08155b7378 ip 00007f08155b7378 sp 00007f08174c26d8 error 15
Next try, tell UI-Installer to build python, too
Everything works and it keeps running, yeah
The difference I noticed, not sure if that has impact, Ubuntu 16.04 uses a newer version of python 2.7.11-1 UI-Installer uses 2.7.10.
It looks like that python2.7 for Ubuntu 16.04 is currently in kind of a bad state, seeing all the posts online ...
So yeah, I guess for the moment it is down to build your own python, or stay with 14.04 and hope the situation will get better or use another OS, if you want/can.
BTW, the UI-Installer can do it for you too, works like a charm ! Thanks to @smcmahon for this !!!