Suggestion for mistake on computation of a pull-back $2$-form on the sphere

The result $f^*(\omega) = 4(dx\wedge dy + dz\wedge dt)$ is correct. This is actually a rather well-known fact, since your $f$ is just the Hopf fibration.

The next step in the calculation is to use the relation $x\, dx + y\, dy + z\, dz + t \, dt = 0$, as you've noted. Specifically, with the $dx\wedge dy$ term, you can convert the coefficient $z^2+t^2$ to $1-x^2-y^2$. The $- x^2$ term can be rewritten using this relation as $$ - x (x\, dx\wedge dy) = x (z\, dz\wedge dy + t\, dt\wedge dy) = - xz\, dy\wedge dz - xt\, dy \wedge dt\ . $$ Similarly the $-y^2$ term equals $yz \, dx\wedge dz + yt\, dx\wedge dt$. You can also work out that the coefficient of the $dz\wedge dt$ term is $4(x^2+y^2)$, to which we can apply the same procedure and generate a few more terms. If you finish computing the coefficients of the other terms (through blood sweat and tears, or alternatively, with Mathematica) in $f^*(\omega)$, you will find that they actually all get cancelled by these extra terms we generated, and what's left will be the desired result.