= Solution
Use algebraic <Chern classes> in the <Chow ring> of a nonsingular <algebraic variety> $Y$. For a <vector bundle> $E$ of rank $r$, the classes $c_i(E)\in\operatorname{CH}^i(Y)$ are invariant under bundle isomorphism, have $c_0(E)=1$, vanish for $i>r$, and commute with pullback. Their <Total Chern class> is $c(E)=\sum_i c_i(E)$. The <Whitney sum formula for Chern classes> applies to every short exact sequence $0\to E'\to E\to E''\to0$:
$$
c(E)=c(E')c(E'').
$$
Normalization for a <line bundle> is $c(\mathcal O_Y(D))=1+[D]$, where $[D]$ is the divisor class. In particular, $c_1(L\otimes M)=c_1(L)+c_1(M)$, $c_1(L^{-1})=-c_1(L)$, and the trivial bundle has total class one. These properties determine the classes: pull back to the iterated <projective bundle> where $E$ has a filtration by <line bundles>. The <projective bundle formula for Chow groups> makes this pullback injective, and Whitney multiplication expresses $c_i(E)$ as the elementary symmetric polynomial of the first classes of the line factors. This also gives $c_i(E^*)=(-1)^ic_i(E)$.
Let $X$ be the smooth degree-$d$ <projective hypersurface> and set $h=c_1(\mathcal O_X(1))$. The <algebraic cotangent bundle> has rank $n-1$. Restriction of the <cotangent Euler sequence in homogeneous coordinates> remains exact, because its terms and quotient are locally free. Thus
$$
0\longrightarrow\Omega^1_{\mathbb P^n}|_X\longrightarrow\mathcal O_X(-1)^{\oplus(n+1)}\longrightarrow\mathcal O_X\longrightarrow0
$$
and Whitney multiplication gives
$$
c(\Omega^1_{\mathbb P^n}|_X)=(1-h)^{n+1}.
$$
The defining equation has degree $d$, so the <conormal sheaf> is $\mathcal O_X(-d)$. Smoothness makes its differential map injective with locally free quotient, giving the <Conormal exact sequence for Kähler differentials>
$$
0\longrightarrow\mathcal O_X(-d)\longrightarrow\Omega^1_{\mathbb P^n}|_X\longrightarrow\Omega^1_X\longrightarrow0.
$$
A second application of the <Whitney sum formula for Chern classes> yields
$$
\boxed{c(\Omega^1_X)=\frac{(1-h)^{n+1}}{1-dh}.}
$$
The inverse is the finite geometric series in the graded <Chow ring>: terms in codimension greater than $n-1$ vanish. Expanding gives every requested class,
$$
\boxed{c_i(\Omega^1_X)=\left(\sum_{j=0}^i(-1)^j\binom{n+1}{j}d^{i-j}\right)h^i\quad(0\le i\le n-1),\qquad c_i=0\ (i>n-1).}
$$
For example,
$$
c_1(\Omega^1_X)=(d-n-1)h,\qquad
c_2(\Omega^1_X)=\left(d^2-(n+1)d+\binom{n+1}2\right)h^2
$$
when the corresponding ranks and codimensions exist. The first expression agrees with the <canonical bundle of a smooth projective hypersurface> $\det\Omega_X^1=\mathcal O_X(d-n-1)$. For a plane curve, $\deg c_1=d(d-3)=2g-2$, matching Q4. For a quartic surface in $\mathbb P^3$, $c_1=0$ and $c_2=6h^2$, whose degree is $24$. These checks also confirm the sign in the cotangent, rather than tangent, formula.
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