Is it allowed to hide the copyright and display it as part of a corporate footer we use on our company website?

You'll need to check with IPS themselves for an official opinion. My recollection is that the copyright can be formatted but must be given at least equal visibility (i.e. no making it smaller or fainter).

and not forgetting to add the link back (not just the text)

as @Meddysong mentioned best to check with IPS themselves if can do what you suggested

You may move the copyright, provided you do not alter its text, alt text, or hotlink back to our site, and also cannot make it less visible. (i.e. you cannot change its contrast to be "dark on dark" and therefore less readable, nor "hide" it in any manner such as behind other elements of the Theme.)

Thanks for your help, I'll make sure I adhere to the above.