BSOD playing Battlefield 3, error 0x00000124

The 0x124 error tends to be caused by hardware problems. You've said you've already tested your GPU with FurMark but 90 degrees seems a little too high. Test your memory with Memtest86+ (if you haven't already) as well as your CPU with Prime 95.
RAM - Test with Memtest86+
Hardware - Stress Test With Prime95

Also have a look at the below post that explains the 0x124 error and some tips on how to fix it.
Stop 0x124 - what it means and what to try

If you have Gigabyte EasyTune 6, uninstall this. It has a history of known problems with Windows 7 as it is only compatible with XP and Vista. Uninstalling has also fixed BSOD's while playing Battlefield: Bad Company 2.

These BSOD's could also be cause by any overclocks you are running. Putting everything down to stock should help. If it doesn't, you might need to up your VCORE as your CPU may need more to keep it stable.

If you are not overclocking, your VCORE should be fine. This being said, run Prime95 and Memtest86 as well as Furmark to see if something is unstable.

I started Prime95 and then left out to do my own business. When I came back (about 2 hours later) the PC was crashed again. Does that mean that some part of my hardware is broken?

I haven't ran Memtest because I have no USB sticks or extra CD's at the moment. But yesterday I ran Windows' own memory test and there were no problems.

The Furmark result I already said earlier, it ran for 5minutes and after that it settled to 90-degree celsius.

Maybe this means that I have to get whole new PC?

UPDATE: Now when I tested out BF3 my FPS is 1-25, it used to be 30-60 (haven't changed any settings). Really looks like my hardware is broken but is it possible that they just break during the night without doing anything? Because couple of days ago I played with no problems and suddenly yesterday morning it just crashed during match, and ever since that the game haven't worked very well. Also, how do I locate which part of my hardware is broken (CPU or GPU or MoBo)?

If Prime95 crashed your computer, I would say that has something to do with either the CPU or RAM. When possible, give your stress test your RAM using Memtest86+ and post back with the results.

After this, the only foolproof way of checking to see which hardware is faulty is replacing them and running Prime95 again. Do you have more than one stick of RAM in your computer? If so, remove one stick and try and play BF3 as well as running Prime95 and Memtest86+. If your computer crashes, replace the stick with the other stick and try again. If it crashes both times, it shouldn't be the RAM. However, it may be both sticks of RAM. In this case, see if you can find any spare memory lying around and put that in the PC and try the stress tests and BF3 once more.

Next, if you have any spare graphics cards try replacing your 5570 with the spare one. If you don't, see if your motherboard has integrated graphics and simply remove the GPU and run the stress tests again. The computer should switch to the integrated graphics and use them instead of the dedicated card.

If neither solution works, we'll see if we can do something with the CPU.

I ran Memtest for 2,5 hours and there were no errors. Time to test another parts now...

UPDATE: Now I ran Prime95 with my old graphics card, same result. After about 2hours the PC crashed again. If there were no problems in GPU or RAM, the problem must be CPU. Right? What should I do now?

Hmmm, well it looks like it. Do you have any other computers?

Yes I think I have another computer.

UPDATE: I tested my CPU with IntelBurnTest couple of times and there were no errors. Maybe the installation of Windows 7 failed or something?

UPDATE2: I installed MSI Afterburner and played Battlefield 3. My GPU temperatures were around 48-degree celsius (not that high in my opinion). But the FPS is just getting worse. It's 1-15 and totally unplayable. It didn't crash at start this time (I left game because of lag after 10min) but there still must be something wrong. The parts of hardware which affects to the FPS must be broken because FPS wasn't used to be that low.