Different schools have different rules as to what they say you can do with regard to applying to other schools. Frankly, and I might take a lot of heat for this, I wouldn’t care what the school says, except that it clearly would be a conflict to apply ED to any other school, and almost no school counselor, who has to sign the form, would allow it anyway. Beyond that, why would a school care if you applied EA everywhere else? It is non-binding and so if you get in at the ED school, it matters not a whit where else you applied or how. If you don’t get in, then you are ahead of the game by having applied EA to some other schools, and why would the ED school that turned you down care about that?
However, to answer your question some schools do instruct you to not apply in very particular ways to other schools. Tulane hasn’t said yet what approach they are taking. All schools let you apply RD to any other school, and most make exceptions for things like rolling admission schools, scholarships that require you to apply EA, etc. In the end, unless someone can point me to a good reason for any other policy, the only moral policy is to just say “no applying to any other schools ED” and leave it at that.
As far as financial aid, that is widely recognized as the only acceptable reason to break an ED agreement, excepting emergencies. No one has ever been sued to enforce an ED contract, and no school could ever claim damages. But to break it for anything other than the above stated reasons would be highly unethical and, some think, might hurt others from that high school that apply ED in the future. Now that last I would doubt unless the reason for breaking it traced back to the school and the GC, such as they allowed two or more ED applications by the same person. But anyway, I think that answers your question.