polo53 said:
PC temperature is fine.
Not enough information for them who cannot see it physically. Do what is suggested.

polo53 said:
Ran Memtest and one stick of RAM came up with errors in one DIMM slot, but not the other. Could this be the issue?
Now follow the "Part 3: If You Have Errors:" of How to Test and Diagnose RAM Issues with Memtest86+

Errors/red lines means one or more RAM is faulty. But the fault may occur due to a faulty DIMM slot, too, which is a motherboard component. Using memtest86+, you can discriminate between a faulty RAM and a faulty motherboard.

How? Say you have two RAM sticks and two DIMM slots. You obtained errors at the test with all RAM sticks installed. Now, remove all the sticks but one. Test it in all the available slots, one by one. Continue the same procedure for all the available sticks.
How to make the inference that is it a RAM issue or it is a motherboard issue? Suppose you have got the result like that:

No code has to be inserted here.
It is a RAM, a bad RAM.

But if you have got a result like that:

No code has to be inserted here.
It is a motherboard issue. The particular slot is bad.

polo53 said:
Added to this I fear my HDD is dying.
Any reason behind believing so other than the sounds?
Do test the HDD using your HDD manufacturer's tool: 7 Free Hard Drive Testing Software Tools

Also, open the computer. Make it sure that the screws holding the HDD with the stand are properly tightened. Also make it sure that the SATA and power are properly connected.