Aha. So the integers that are squared to get the magic square of squares are required to be neither consecutive nor distinct.

It would be useful if this question defined what a "magic square of squares" is. In any case this appears to address only the 3x3 version.

No. One can find a smooth homeomorphism of the open disk that takes each circle about the origin, say of radius r ≥ 1/2, to itself by a rotation by angle 𝜃(r), satisfying 𝜃(r) → ∞ as r → ∞. This cannot be extended to the closed disk.

I find it beyond ridiculous that this question should be closed. It is a perfectly reasonable question, since it is about an important unsolved problem in research mathematics. There is nothing gained by closing such a question. In fact, it led to Jochen Glueck's very useful debunking of the paper in his answer below.

Maybe if you reread my comments you will be able to clarify your question.

It would be better to edit the question for clarity. But again, I have no idea what you mean by a curve that it "can roll down". Almost any down-sloping curve is one that almost any shape "can roll down".

Also: Where you write "What I find more interesting, are shapes that actually have a different outside boundary" I have no idea what you mean. Any two non-congruent shapes have a "different outside boundary".

"In a smooth manner" is rather vague. Must the shape's center of gravity remain at the same y-value during the kind of rolling that you prefer?

Since each of "conjectures' 1., 2., 3. are multi-line, there is no way to know what "the entire line" refers to. A much better idea would be for you to edit the question so that it is unambiguous.

It is hard to know when the "conjectures" you mention end and the next phrase (if any) begins.