= Solution
Use the paper's <Fubini-Study form> normalization $\widetilde\omega_{FS}=\frac{i}{2}\partial\overline\partial\log(1+|z|^2)$, the <Hamiltonian vector field> convention $\iota_{X_H}\omega=dH$, and the <Lie algebra> generator whose <circle> action is $e^{i\theta}$. Write $D=|z_0|^2+|z_1|^2+|z_2|^2$. The normalized <moment map> for the <torus> action is
$$
\boxed{\phi([z_0,z_1,z_2])=-\frac1{2D}\bigl(|z_1|^2,|z_2|^2\bigr).}
$$
The <moment map> image is the filled triangle
$$
\boxed{\phi(\mathbb{CP}^2)=\{(u,v):u\leq0,\ v\leq0,\ u+v\geq-\tfrac12\}.}
$$
Indeed, the three quantities $|z_j|^2/D$ are nonnegative and add to one, and any such triple is realized by choosing their square roots as <homogeneous coordinates>.
For the diagonal <circle> action, a point in the <fixed-point set> satisfies $[z_0,tz_1,tz_2]=[z_0,z_1,z_2]$ for every $t\in S^1$. If $z_0\ne0$, the projective scaling must be one, forcing $z_1=z_2=0$. If $z_0=0$, all coordinates are scaled together. Therefore
$$
\boxed{\operatorname{Fix}(S^1)=\{[1,0,0]\}\ \sqcup\ \{[0,z_1,z_2]\}\cong\{\mathrm{point}\}\sqcup\mathbb{CP}^1.}
$$
By part (b), the diagonal <moment map> is
$$
\boxed{\mu([z])=-\frac{|z_1|^2+|z_2|^2}{2D},\qquad\mu(\mathbb{CP}^2)=[-\tfrac12,0].}
$$
The factor $1/2$ follows from the specified <Fubini-Study form>, not the integral normalization in the linked general article. For example, in one affine complex coordinate $w=re^{i\theta}$ the <symplectic form> is $r(1+r^2)^{-2}dr\wedge d\theta$, whose <interior product of a differential form> with $\partial_\theta$ is $-r(1+r^2)^{-2}dr=d[-r^2/(2(1+r^2))]$. Choosing the generator $e^{2\pi it}$ rescales all <moment maps> by $2\pi$; reversing the defining sign reverses their signs. The negative level in parts (d) and (e) uses the convention displayed here.
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