Question - even if I bypass the disc integrity check, will it still give me issues and even though the original disc maybe bad, if I copy the image and re burn to a new DVD, will it still fail ?
I'd say yes. I'm not sure what all the integrity check looks at, but every time I've ignored a failure and installed anyway, there have been problems. Some don't manifest right away, sometimes the machine boots just fine, but inevitably - and quickly - it becomes unstable. Could be on updating, or on opening a particular application, or something else; but
every single time I've ignored a failed disc integrity check, and managed to complete an install, the OS was corrupted or compromised in some way.
I don't understand how a good installation file/image/whatever, on a good CD or DVD, somehow just "goes bad." But it happens. I've even had this happen on official Apple install media. One day they pass the integrity check, and the next, they don't. And they don't "recover," they're never any good after that IME. I've been through that over and over, until I finally learned
not to skip the integrity check before installing and
not to try to install from a disc that fails the check.
When you think about it, it makes sense - Apple must have included that step for a reason, they had to know discs could fail, and they didn't want people blaming their OS.
EDIT: I didn't address your other question. I don't think that copying the image from a disc that fails integrity check is going to do anything other than waste your time. It's not the physical media that's the problem - that may have contributed to the problem, but it's not the primary problem any longer. The real problem is that the bits in the install image have somehow become rearranged. The image itself is corrupted, so copying it onto different media isn't going to cure that - it's just going to copy the problems.
I've mentioned before that I used to work in law enforcement. When digital photography became a thing, we all went to it for convenience, but pretty quickly the corruption of .jpg/.jpeg files reared its ugly head. We tried different forms of storage to preserve photos for evidence, and what we found was that the best method was to store the files on hard drives. Something about the magnetic media on an actual hard disk preserves files better. Flash media, floppies, and CD's just had too high a failure rate, no matter what steps we took to shield them from magnetic corruption from external sources. We've since largely moved on from .jpg/.jpeg format to others that seem to be less likely to break over time, .tiff for instance. As storage has become cheaper, we learned that the best practice was to maintain RAW image files.
But I digress. My point was, any data can become corrupted, even on an optical disc. So no, I wouldn't bother copying the problems, I'd just throw out the disc, find another source and start over. That is all.