Also question is, how do I know if my CPU fan is going bad?
Look and Listen. The fastest way to tell if your computer's fan has stopped working is to visually inspect it. If it's a desktop computer, you can usually see the fan rotating on the back of the case. If the fan is not moving, you must replace it immediately, before using your computer further.
Additionally, should I replace my CPU cooler? A new CPU cooler alone will do okay but still you need airflow going in and out of the case, if you want your machine to overheat to save a buck well thats up to you. Sticking with the stock cooler will be fine if you do this, i mean its a non overclocking core i5 so no need to get the CPU cooler if you get case fans.
Just so, how long should a CPU fan last?
Anywhere between 0.001 seconds and 200+ years. As with any manufactured product, it all comes down to 40% manufacturing and 60% luck. I've been running this pack of 120 mm Rosewill fans 24/7 for about the last 8 years without issue. I also bought an Alpine CPU cooler whose fan didn't list much more than a year.
Can CPU work without fan?
Todays processors create so much waste heat that it will only take a few seconds for it to go into thermal shut down. Running a CPU without a fan and heatsink is a sure fire way to fry your CPU and when it's fried you can argue with Intel all you want but they aren't going to replace it.