Is copying a lot of files bad for the CPU or computer in any way?
Damage to CPU – Minimal, if any
1a. CPUs are not heavily involved in file copies. The CPU's job is to tell the other hardware to do the transfer, and the actual transfer of 1s and 0s is left to other chipsets/IOs to do.
1b. Even if a file transfer did use a lot of CPU, CPUs have incredibly good longevity and usually processors don't fail or go bad. They do get replaced regularly due to being outdated, which speaks to the fact of its longevity – a CPU will outlive its efficiency lifespan.
Damage to HDD – Still minimal, but worth considering
Using an HDD wears an HDD. It has moving parts, so the more they're used, the more potential for damage from wear – but if you can't use it, it's useless anyway. The best practice for HDDs is to know that at some point, it will fail and to have backups. This is the only real protection you have against HDD failure: backups.
You can reduce the wear and tear on the HDD by ensuring that you run a single copy command at any given time (which is what I suspect is the typical method), so whether you do a drag & drop or run a command like
robocopy, make sure you only do one of these actions and let it complete before starting another one.
Your specific questions
No, you will not affect your CPU in a negative way. In fact, you can open task manager during this and probably witness your CPU not go over 20% usage, depending on what else your computer is doing in the background.
No, all CPUs are built to be robust. A 3rd gen i7 vs an 8th gen i9, or a Ryzen7, what have you, is engineered to last and endure utilization far above what a file transfer will do to it.
No, this would not be a helpful method to reduce the damage that we've already stated is negligible. Even if your CPU isn't properly cooled, this low utilization should not make it overheat – and if it does, its performance will only be affected while it's too hot. Its lifetime performance/longevity will not be affected by overheating.
No, for two main reasons.
Intel and AMD CPUs in desktop and laptop computers, if properly installed, cooled, not overclocked, and not subject to electrical problems like a bad power supply, will last a very long time even if all cores are constantly 100% utilized--such that you don't usually have to worry about "wearing out" your CPU.
Furthermore, even from the first PC's of the early 80's, disk drives used something called Direct Memory Access (DMA). Floppy drives have used DMA since the first PC, and hard drives started to have DMA modes probably sometime in the late 80's.
DMA basically allows drives to transfer data to and from RAM without the CPU in the middle. So hard drive transfers have even less of a CPU impact than you might think.
Where the CPU might still be in the middle is if you have disk encryption running on your system, such as Bitlocker, TrueCrypt/VeraCrypt, or other. But it's still not going to be anything to worry about.
You are causing an HDD hard drive (not an SSD) to work harder by copying from it constantly. You are adding wear to an SDD by copying to it. Spinning-rust HDDs and even SSDs do have a finite lifetime that is likely less than your CPU.
TL;DR - you should be much more worried about the lifetime of your drives (and have backups/multiple copies) than the CPU.
Most hard disks are rated for a few terabytes read/write per day for several years. Copying 20 TB once will use up a few days of the disk's lifetime, which is counted in years, usually 5-10 years.
You may assure yourself that this is true for your disks by consulting the manufacturer's specifications.
As regarding the CPU, such an operation will not use it at all. The CPU will most of the time be waiting upon the disks, so will not even warm up noticeably.
So don't worry and go ahead. If something fails during the operation, this only means that the component was already in bad shape and would have failed anyway in a few days time.