Is the fan in my MacBook Pro broken?

Yes.

Unless you have something that got into your CPU fan module as pointed out, the noise is probably (in my experience) from the bearings in your CPU fan being worn out and your CPU fan module would need to be replaced. This would come and go as you describe. If tapping moderately on your MacBook gets the fans to stop rattling, this lends more support that it is the bearings in the CPU fan.

Also, if you using Mac Fan Control to ramp up the RPMs to max doesn't quickly cool down your CPU (especially if you are not watching video) this is a sign of something going on. You should be able to ramp up the RPM to max, hear the fans spinning much more audible, and see the temperature drop.

Usually, it's just really annoying to have a noisy fan, but since your temperature is getting critically hot it needs to be fixed right away. As pointed out, 80-degrees C is a fairly good comfortable maximum.

The good news is that it's an easy fix and a $20 part:

https://www.ifixit.com/Guide/MacBook+Pro+13-Inch+Retina+Display+Early+2015+Fan+Replacement/45131

Hi Fromthemountain,

Thanks for the post in Apple Support Communities. It sounds like you're running into some issues with the fan on your MacBook Pro.

Have you tried to reset the System Management Controller? This handles thermal management, among other things: How to reset the System Management Controller (SMC) on your Mac

Also, you mentioned the Mac slowing down. You might check Activity Monitor to see if there's an app using significant resources as shown here: See how apps affect Mac performance, battery runtime, temperature, and fan activity. This might give you an idea as to whether there's a software cause for some of the behaviors you're experiencing.

Regarding the fan issue, the SMC reset might be useful. If you were hearing a rattling sound and your fans don't seem to start up after the reset, I suggest getting in touch with Apple directly to look at your options: Contact Apple Support

Take care.

There is nothing built into your Mac that could cause rattling. Only a foreign object would produce that issue.

It now sounds like whatever that foreign object is has lodged itself in the fan blades and stopped the fan. 95 degrees is very near emergency-power-off fail-safe temperature.

You should take your Mac in to an appointment at an Apple owned store for a free evaluation.

If you have only an Apple-Authorized Service Provider available, they may charge a nominal fee for the evaluation, sometimes waived if you get a paid (or warranty) repair.

Oh, I didn't see the audio/video you posted until just now.

Based on that sound I have to agree that's much more likely something caught in the fan module (not worn out bearings I don't as I said above). One can literally hear the fan blades smacking into something. A can of compressed air might do the trick with the fan exhaust facing downward such that whatever is in there drop out. Although it may be part of the fan itself broke.

In or out of warranty you can get a free over the counter 'Apple Service Diagnostics' test /assessment

Make an appointment for a "hardware issue"—

https://www.apple.com/retail/geniusbar/

From iOS app online Genius Bar appointment:

https://itunes.apple.com/us/app/apple-support/id1130498044?mt=8

Outside the USA

https://locate.apple.com/country