CPU Fan and GPU Fan Acting Funny?

So you need to replace the fan in your CPU cooling system. That does NOT mean you have to replace the entire CPU cooler. Most such systems (IF you have a fan-only system and not a liquid cooler system) consist of a heatsink with fins mounted firmly on top of the CPU chip, and a fan attached to the heatsink to blow air though the gaps between fins. For most of these the fan is a "standard" computer fan (often 80mm or 120 mm size), and you can replace the fan only. To do that you need to take these steps
1. Examine the fan label and identify it. You need to know especially its maker and model number. Look that up on the web and find its air flow rating and its pressure rating. Fans for CPU heatsinks are usually labeled as "high pressure" units because they must blow the air through more backpressure than a case fan. IF you can get an exact replacement (either though a store or from the CPU cooler maker) that is ideal. Otherwise look for a fan of the same size with airflow and pressure ratings close to the old one, or slightly higher. Also take note of the number of wires from the fan, and thus of the number of holes in the connector on the end of the wires (3 or 4). Try to get a replacement with the same.
2. Examine closely how the fan is fastened to the heatsink. Some use metal wire clips to hold it on. Others simply fit screws through the fan chassis' corner holes and into the spaces between fins of the heatsink. Use that as a guide to how to un-attach the fan and then re-attach its replacement.
3. You do NOT need to remove the heatsink part from the CPU. Just remove the fan from the heatsink, then replace it. Try hard NOT to put any stain on the heatsink and CPU.
4. When you are doing the replacement and while the old one is off, examine the slots between the fins of the heatsink. Often they have an accumulation of dust in them that stopes air flow. If yours does, clean them out. You may need to use a small thin stick to loosen the dust so it can be blown out. After they are clean, install the new fan.
5. Warning: When cleaning dust, do NOT use a vacuum to pull air and dust through a fan without holding the fan blades stationary. A big flow of air through a free fan can cause it to turn MUCH faster than it is designed for and damage the fan's bearings.

Now on to the GPU fan. The spec you gave, "Nvidia GTX 1050" is not enough. Nvidia makes GPU major components but sells few video cards directly. Most such cards actually are designed and made by others like ASUS, EVGA, Gigabyte, etc. So tell us more about the card's markings. What I suspect - and that's why I need more info, to check this idea - is that the video card controls its own fan speed and is set up to NOT use the fan when the GPU chip on it is still very cool from not being in use. In that system, the fan might start briefly to ensure it works, then stop while the card is cold. Later it will start up slowly, then speed up more as you use it. But in your current situation with a CPU that has NO cooling and thus shuts down the whole system, you never get to run it long enough for the video card to heat up and require fan cooling. In other works, the GPU card fan may be just fine, and will work perfectly well once you solve the CPU chip fan problem.