I believe Bing used the term pseudo-isotopy in a different sense from what you have in mind. In Bing's sense a pseudo-isotopy is a 1-parameter family of maps $f_t$, $0\leq t\leq 1$, which are homeomorphisms for $t<1$ but with $f_1$ allowed to collapse nontrivial subspaces to points.

The unpublished paper of mine that Sam Need refers to is "Bianchi orbifolds of small discriminant", available on the arXiv. It includes six examples of hyperbolic orbifolds homeomorphic to ${\mathbb R}^3$, the quotients ${\mathbb H}^3/PGL(2,O_d)$ for $O_d$ the ring of integers in the quadratic extension of ${\mathbb Q}$ of discriminant $d$ for $d=-3,-4,-7,-8,-11,-19$. These examples rely on computer calculations by Robert Riley from the 1970s. In each case the singular locus of the orbifold contains points with noncyclic stabilizers, so these examples are different from Sam Need's example.

The long exact sequence should have a term $H_0(T^2)$ between $H_1(Y)$ and the final $0$. Without this term the sequence will not give the correct calculation of $H_1(T^3)$ for example. The extra term arises from the fact that $\phi_*-I_* : H_0(T^2)\to H_0(T^2)$ is the zero map.

@EthanDlugie: I was referring to your statement that a set is open if it is open in each cell. If this were true with the usual definition of cells then every subcomplex would be an open set. For a set to be open it needs to be open in the closure of each cell.

The argument in this answer seems to be using a nonstandard definition of what a cell in a CW complex is. According to the standard definition, as in Whitehead's original paper and most recent sources, a CW complex is the disjoint union of its cells, so cells are "open cells". Unfortunately some authors in the intervening years took "cell" to mean the closure of what Whitehead called a cell. With this definition a "cell" need not be contractible. For example, every smooth closed connected manifold is a "cell" by this definition.

It is a theorem of Livesay (1963 Annals of Math., pp. 582-593) that $M=P^2\times I$ if $N=S^2\times I$. A shorter proof was later published by Rubinstein in the 1976 Proceedings of the AMS, pp. 317-320.

The Hopf invariant 1 problem was first solved by Adams. A later simpler proof is due to Adams and Atiyah. The Adams Conjecture (which is not needed for the Hopf invariant 1 problem) was first proved by Quillen, with subsequent proofs by Sullivan and Becker-Gottlieb.