I realize these are hypotheticals, but now you might be edging into where it is easy to be asking “permission” to do something unethical. But the problem is that the parameters of your hypothetical are incomplete. It depends on your actual financial situation. So if the reason you got no aid at Tulane is because the numbers show your family can afford the full cost of attendance AND the student did not qualify for merit scholarships, then it would still be highly unethical to apply and get in ED and break the agreement because School Y came along with a full tuition or full ride offer. Bottom line: If the idea of attending a school for free is that attractive to you, presumably a much less competitive school than Tulane if they are offering a full ride when Tulane offered nothing, then applying ED is not the option to take. I think it is fine to use the financial calculators and other knowledge to reasonably predict what you are likely to receive and to use that to say that would make Tulane the choice no matter what any other school offers. In that case ED is perfect for you. But applying ED to Tulane to get an edge getting in if the plan all along is to abandon Tulane if a full ride offer comes in, no matter your financial circumstances, would just be wrong.
I am just expounding on that hypothetical, not saying that you are contemplating any of this. It is just that the incompleteness of your question leaves a lot of possibilities open. Too many really. Applying ED to any school means you should attend there if accepted except under the most extraordinary of circumstances.
As to your last questions, it is possible but unlikely a school you broke an ED agreement with would even know where else you got in, but even if they did I am not sure they are allowed to contact them with the kind of information you are talking about. But it should be totally moot and not worth discussing, because (and I know I am repeating myself but it is that important) unless you truly cannot afford the ED school, then there is no reason to consider School Y’s offer at all, no matter how tempting. This should be totally thought through before applying ED anywhere. If you know you need $X to afford Tulane or another school ED, then you should do your homework as to how likely you are to receive that amount before pulling the ED trigger. If you did and then that money doesn’t materialize, you are morally in the clear. But it can’t be just a made up number that Tulane never was likely to give you. It has to be based on facts.