You don't 'need a heat sink for the drive. It's just if you plan on benching it constantly because they will hit 100C when you hit them with constant data for a prolonged period of time. My SN850 sits around 40C in normal operations. If you have proper airflow it's usually not an issue to keep it stable / cool / throttled. I use the same in my laptop and it stays a bit warmer ~45-50C due to the compact nature of laptops. I also use a SN770 as a secondary drive in the laptop and it does the same for temps.
As to the HDD's though... 12 years is a bit much for my comfort but, if it's not spitting out errors and data isn't becoming corrupted it's just something to keep an eye on if there's data you can't lose. If it's not a huge drive then pairing it up with another one of the same size and mirroring it would be an ounce of prevention to keep the data safe for when it does die.
Just keep in mind that some newer boards will require switching the boot drive to GPT due to UEFI. It's a simple process to move the data around and convert over but, likely a better idea to do a fresh install of the OS since it's a new drive to be used anyway. As long as the mobo is set to UEFI the drive / OS install will automatically do this for you in the proper format.
Since the HDD's listed on PCP are only 1TB/ea picking up a 4TB shouldn't be a huge leap to consolidate things and expand your storage. I would actually go higher to avoid the issues with some drives using SMR which slows things down a bit when you need to re-write the data.
Comparison of all hard drives and SSDs on Amazon, sorted by price per TB
Since you picked an AIO the cooling issue on the NVME will be less because of the CPU heat being diverted completely through the loop instead of an air tower. Though you could use an air tower for $50 and put the other $125 into something else. I use a SE-226-XT and a graphite pad instead of paste and keep my 12700K at room temp 99% of the time it's running. This might go up when you're gaming but, generally LC isn't really needed even though it's hyped to the extreme all over the internet.
As to the case, that's just personal preference. I went a bit more subdued with a Meshify 2 from Fractal and it's got tons of fans inside to push air. It has room for 6 x 140 / 1 x 120 and even more if you get creative with mounting. As for the fans I'm using the Arctic P12/P14 PWM PST because you can chain them together on a single header to keep wiring out of the way. Also, they allow me to put 2 of them on the cooler for the CPU using the single CPU header.