Can original file get corrupted during copying?

Is there any chance that the original files on my computer could get corrupted because copying process failed for any reason?

Any chance? Yes, but it's more likely that you get struck by lightning while being bitten by a shark... Theoretically it could happen, but the chance is so low it is realistically impossible as the original file is only read and not written to. That does not mean some other thing couldn't corrupt a file while the copy was in process, such as a failing HDD, RAM, or being read/written by another process.

What would happen if I would be moving the files instead of copying?

When moving a file, a copy process occurs first, then when the copy is considered to be successful, the original file is deleted. It doesn't copy 12% of the file, fail, then delete 12% of the original file. It is more likely for corruption to occur doing this, but not significantly more.

Honestly, I would look into TeraCopy or similar utilities... It is a replacement for Window's copy utility which has the added ability to verify the destination file(s) after a copy process.

EDIT: Added for clarity to the question being updated.

The copy process varies slightly based on the application, but it is essentially read into RAM in chunks of varying size, then written to the destination file in chunks until the process is complete. How big the "blocks" are varies by application, file size, available RAM, etc. In a move processes, it also varies by application, but most leave the partial/failed file at the destination but leave the source file untouched. Note that when files are moved within the same partition, most modern operating systems are smart enough to just update it's "location" metadata, rather than actually move the data. This method can be significantly faster and more reliable than an actual copy/delete process.

Is there any chance that the original files on my computer could get corrupted

There is a small chance that files could get corrupted at any time (not just while copying).

To mitigate the risks you can do things like make sure you are using ECC memory and also perform checksums before and after a copy (or do a complete bitwise compare of the two copies).

There is an interesting discussion here, statistics - Cosmic Rays: what is the probability they will affect a program? - Stack Overflow, where the answers discuss the probability of random corruption and what can be done to prevent it.

Soft error - Wikipedia is also worth a read.

The chance that the original file would be corrupted, any more than any other read operation would corrupt a file, is infinitesimally small. Most file corruption happens during a write, and is usually combined with a power or hardware issue. Same thing with a move, which is actually a copy-check-delete process.