Well, because of that you noticed there could be two options:
1st: Your fan is defect
In my opinion the most logical idea regarding your observations, especially after the buzzing is not present the whole time and got worse if you stream something (= the system is more under pressure which causes a faster fan speed). If the fan is going to die mostly the system gets hotter due to the missing cooling and has no option to cool it down - this ends up in a slower system and finally freezing / sudden shut downs. In this case you also should notice a clear heat up of your system overall.
2nd: Your HDD is defect
Also the HDD could make that sound, but - if the system runs fairly normally under less load - I don't think it's the issue here. A defect HDD is shown in a overall slow system, sudden and sporadic freezes or finally no boot (HDD not detected).