The Wikipedia universal enveloping algebra suggest a way to obtain higher Casimir operators (e.g. generators of the center of $\mathfrak{U(g)}$ for $\mathfrak{g}$ semisimple) by evaluating certain determinant:

$$ \det\, (t\mathrm{I} - \mathrm{ad}_X) = \sum_{i=0}^{\dim \mathfrak{g}} p_i(X) t^i $$

However if one does that for $\mathfrak{sl}_3$ then the resulting polynomial has only degrees (in $t$) 0,1,2,4,6,8 and it's coefficients seem to be just powers of the quadratic Casimir operator. If one tries to do the same for the defining representation $\mathbb{C}^3$ (replacing $\mathrm{ad}_X$ by $\rho(X)$ in the above formula) then one obtains quadratic as well as cubic invariant polynomials.

In the proof of Harish-Chandra isomorphism as presented e.g. in (1) there is construction of elements of $Z(\mathfrak{U(g)}$ using traces of matrices from representations of $\mathfrak{g}$. Something like $$ \sum\mathrm{tr}(\rho(X_{i_1})\rho(X_{i_2})\ldots \rho(X_{i_n}))X_{i_1}^*X_{i_2}^*\ldots X_{i_n}^* $$ where $X_i$ form basis for $\mathfrak{g}$ and $X_i^*$ form dual basis with respect to Killing form.

Q1: What is going on here?

Q2: Is it true that for a semi-simple complex Lie algebra and it's smallest nontrivial representation one obtains in this way all generators of the $Z(\mathfrak{U(g)}$?

Q3: Does the approach through determinant give the same operators that appear in the proof of the H-Ch isomorphism?

(1) *Cohomological Induction and Unitary Representations* by Knapp, Vogan