MBPr with El Capitan freezes at or right after login
So it's two months later and I'm a little bit wiser and would like to summarise what I've already posted in the comments and expand on that little.

What my colleague forgot to tell me was that the computer had been freezing from time to time even when running Mavericks; he only mentioned freezing after upgrading to El Capitan, which led me to believe that software was probably to blame. It turned out that after upgrading the freezes started to happen more often.

The laptop was sent to a repair shop and the technician from there told me he discovered traces of liquid at the inside of the cover. Not enough to damage the humidity indicators which are placed inside Apple products but he said there were clear marks so the computer probably had something spilled on it - too little to wreck it immediately but too much not to have an impact on sensitive electronic parts.

Answering my own questions:


The MacBook was submitted to a repair shop where a technician diagnosed the problem as some kind of motherboard failure but he was unable to pinpoint which element was responsible for the issue. Since he wasn't sure which part he should target for replacement he suggested replacing the whole motherboard.
No known solutions to the problem came into evidence.
I was able to download Mavericks from AppStore by finding it in purchase history, as suggested by Tetsujin (thanks!). However, it turned out that software was not responsible here -- after installing Mavericks the computer hung just as willingly as it did under El Capitan.
There wasn't much more to be done before submitting the laptop to a repair shop beyond what I had done and described in the OP.


So if you're experiencing similar problems, it's likely to be a hardware failure, in which case you won't be able to easily diagnose it on your own and you're bound to visit a repair shop.