MBPr with El Capitan freezes at or right after login

Here's the problem: my colleague has a MacBook Pro Retina with El Capitan installed and he's been having issues with regular freezes lately. Everything has worked fine under Mavericks for well over a year, then it was upgraded to El Capitan, worked mostly fine for some time and then started to hang up regularly.

Freezes happen on/during either of the following:

• the boot screen
• the login screen (most often)
• within seconds after logging in

Freezes do not happen in either single user or safe mode.

As stated above, the machine hangs most often at the login screen. I start typing the password and bang, after a few characters it just stops responding - I cannot type anything on the keyboard or move the pointer with the touchpad. Keyboard backlight is on permanently and doesn't turn off.

After booting into safe mode in /var/log/system.log I've found lots of entries like this:

WindowServer[168]: GPU Driver appears to be hung (over 5 continuous seconds of unreadiness)
--- last message repeated 817 times ---

The problem is reproducible on all users. Since we have a Time Machine backup and I could experiment freely, I've tried reinstalling El Capitan (keeping user files). No change. Then I wiped the system drive and did a fresh installation of El Capitan, created a user during the initial setup and crashed into the same problem, before even restoring the backup.

Hardware test (accessible by holding S during boot) was run twice and did not yield any results.

The disk was encrypted with FileVault before my wiping it clean. Now it's unencrypted and the problem persists, so this doesn't seem to be of relevance.

My questions here are:

1. What could be a possible cause of such problems?
2. Is there any known solution to these?
3. Is it possible to download Mavericks from a reliable source and installing it over El Capitan? That's the last OS X version known to work well on this machine.
4. What can I do to debug the causes?

Any suggestions not contained in the questions are also welcome.

Solution 1:

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:

1. 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.
2. No known solutions to the problem came into evidence.
3. 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.
4. 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.