Code is protected by copyright as a literary work. The method is not protected by copyright, that would be the domain of patents. What's protected are the words.
If you say "Claude, build me a website about X" then you do not have any creative control over the literary work Claude is producing. You just told a machine to write it for you. Nor, like a compiler, is it derivative of any other work that you wrote.
If, on the other hand, you are working jointly with Claude to make specific changes to the code on a line-by-line basis, then you will have no problem claiming copyright over the code. Claude in this case is acting as a tool, but there's still a human making decisions about the code.
In the case where you wrote a bunch of markdown and then told Claude to generate the corresponding code but didn't have any involvement in writing the code itself, you could perhaps claim that the code is a derivative work of the markdown, a court would have to handle that case-by-case basis and evaluate how much control you exerted over the work.
No, a copyright application can be filed with a corporation listed as the author. Watch for the copyright notice at the end of the next major movie you see.
In any case, the corporation did not create the product, people created it and their contractual relationship with the corporation defined how the ownership of that work was managed. So, I don't find it too unusual that this element of personhood is available to corporations.
Under at least EU AI Act, any work done by AI is not granted copyright. But it does not mean copyright does not apply, it means the amount of work credited to AI is set at 0% (simplification). A human working off another's work unless it's perfect copy will have "credit" for changes that are judged creative/transformative, meaning a human plagiarizing something still can claim to have some degree of authorship. An AI won't.
In a sense, the copyright status of final work is a sort of "sum with dilution" were each work involved adds to claims, but AI's output is set at 0 - the prompt or further rework by human is not.
As for employer, details vary but generally "work for hire" rules and contracts do reassignment of material rights (in EU and some other places you can not reassign moral rights which are a different thing).
I think what this means is that the employee may not be the copyright owner for multiple reasons, which are possibly applicable simultaneously. It does not imply that the employer owns copyright over the work that is in public domain, which would be a contradiction.