After a while of having the Mac Mini running half the way, with an unusable video card, it died again.
I had planned before the whole disaster a trip to New York (we don't have Apple Stores at my country).
I was planning on buying a used Logic Board so I could replace the board and hence the video card. But it's expensive and a hard piece to find.
The price range was between 320 and 400usd. But only from private sellers in Amazon.
So I took my Mac Mini to the Apple Store, I had nothing to lose (they consider the RAM and disc upgrade as user replacement so they didn't care that it wasn't the factory model). I had a meeting at the so called Genius Bar, we ran all the possible tests, we couldn't find the problem. They advised me to check in the mac in order to get a better diagnose, since they don't charge for that, I accepted.
After a few days, they told that they couldn't precise what the problem was, but they were pretty sure that by changing the logic board, it could be fixed. They would charge me 400usd for the replacement and labor (only 39usd of labor, the rest the replacement itself). I stated that if the new piece would have the same video card I wouldn't be quite satisfied, they told me that from what I told them and what they found on the internet they already had a different video card on the logic board (I didn't quite believe the guy at that time). Anyways I accepted.
Another few days went by, I called again and they told me that they couldn't fix the problem with the new logic board, they did some new tests and something went wrong. So they were going to replace the computer, I asked if I could get my SSD back and they complied.
I went to the store to pick up the new computer (by that time I didn't know if I had to pay anything or not, nor which model I was getting).
They were going to give a new Mac Mini (late 2014 I think), the same gamma I had, the middle one, free of charge. Since I needed an SSD for a better speed for my work I upgraded it to the top model. I payed 326usd tax included and I got a new Mac Mini late 2014.
I'm quite satisfied with the results.
If you have your mac minis just sitting there I recommend that you take them to an Apple Store and see what happens.
Hope this helps!