Business registration is a matter of public record, as far as I'm aware. Any entity you might set up for tax purposes would need at least to have a registered public agent, and probably a public list of directors and shareholders. However, it would mean that the business you're contracting with to no longer needs to file a 1099, at least.

No, if you (as a contractor or employee) make more than $600 from the same person/business, you'd be obligated to report it, which would in turn make them obligated to report it, even if they didn't realize you had done more than a single contract for them. For this to work, you'd have to have independent shell corps make each chunk of work as well, which .. is probably also illegal.

What could work is a middle man (ala upwork), but then you'd have to accept the middle man knowing you.

you'd have to spin up an LLC for each engagement to do this, 1099 forms are based on your entire (contractor) relationship with the company for the tax year.