= Solution
Let $I(\kappa)$, $W(\kappa)$, and $M(\kappa)$ mean respectively that $\kappa$ is inaccessible, weakly compact, and measurable. Define a fourth cardinal property
$$
\Theta(\kappa)
\quad\Longleftrightarrow\quad
\bigl(W\mathbf C\land M(\kappa)\bigr)
\lor
\bigl(\neg W\mathbf C\land I(\kappa)\bigr).
$$
Take $\Phi_0=I$, $\Phi_1=W$, and $\Phi_2=\Theta$.
If inaccessible and weakly compact cardinals exist, Question 1a shows that an inaccessible lies below every weakly compact cardinal. Thus $I<_1W$. If $W\mathbf C$ and $\Theta\mathbf C$ both hold, then $\Theta$ is exactly measurability. Every measurable cardinal is weakly compact, and the usual ultrapower reflection theorem gives weakly compact cardinals below every measurable cardinal. Therefore $W<_1\Theta$.
Now assume the consistency of ZFC with an inaccessible cardinal but no weakly compact cardinal. In such a model $\Theta(\kappa)$ is exactly $I(\kappa)$, so
$$
\iota_\Theta=\iota_I.
$$
Consequently $I\not<_1\Theta$. This is an explicit <nontransitivity of the least-occurrence order on cardinal properties>, even though $I<_1W$ and $W<_1\Theta$.
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