1. I check them first then copy and paste and check them again.
2. If that file is corrupted I delete that file and do the copy and paste. If that dose not work that means one or two things. The disk drive that I put file is broken or the cable needs replacing. I leave that file where I put it in the first place until I get a new disk drive or get that cable replace. With the bad disk hard drive I get a new one and hope those files are fine. Then I get someone with the right software to transfer the file from one drive to the other. I know it cost much but it will be worth it. If the hard disk drive that where I put my back up file are fine by clicking one of the other file and its fine well its the cable and I get it replace.
3. I don't trust 3rd party programs. Like I don't trust mod managers to install my mods for games. I do it manually. All files comes in different names. I keep my files whilst others deletes them. When I get time I put them in folders. In those folders there are 20 different files just in case one file is better then the other.
I can understand not trusting software to do automatic backups, but to not trust backup software period doesn't make any sense. It's nothing like mod managers at all. Backup software can do things not easily done by a human in that it can run checksums of your files to guarantee integrity (though not all backup software does this). I don't know how many files you are talking about, but doing a manual check of each file isn't feasible when talking about a bunch of files, but for software they can do it by just comparing the old to the new and then maintaining both original and changed copies (something you don't seem to be doing with your current backup, this is known as versioning) meaning if somehow the software does goof, you can just use the older version.
That said, if you are set on your current method, I'd suggest looking into ZFS for your backup's file system. ZFS can do automatic checksums and fight off bitrot, so would perfectly compliment your current backup strategy and doesn't involve the use of software (ZFS is a filesystem like NTFS or FAT, in case you didn't know). Granted, to do so you'd need to set up a NAS, most likely (FreeNAS can do ZFS).