= Solution
For a real or <complex vector bundle> $E$, choose a fiber metric. Its <Thom space> is the pointed quotient
$$
\operatorname{Th}(E)=D(E)/S(E),
$$
collapsing the <sphere bundle> to one basepoint. When the base is compact, this is the <one-point compactification> of the total space: fiberwise radial compression identifies the total space with the interior of $D(E)$, and all directions approaching its boundary become the single point at infinity.
Decompose $\mathbb C^{n+m+2}=V\oplus W$ with $V=\mathbb C^{n+1}$, $W=\mathbb C^{m+1}$, so the subspace being collapsed is $\mathbb P(V)=\mathbb{CP}^n$. Every line outside it has a representative $(v,w)$ with $w\ne0$. Its projection to $W$ gives the line $\ell=\mathbb Cw\in\mathbb P(W)=\mathbb{CP}^m$, and there is a unique complex-linear map $A:\ell\to V$ such that $A(w)=v$. The line in $V\oplus W$ is precisely the graph of $A$. This construction is unaffected by replacing $(v,w)$ with $(\lambda v,\lambda w)$.
Let $\gamma=\eta_1$ be the tautological line over $\mathbb P(W)$. The graph construction and its inverse are continuous, indeed smooth in the usual affine charts, and identify
$$
\mathbb{CP}^{n+m+1}\setminus\mathbb{CP}^{n}
\cong\operatorname{Tot}\bigl(\operatorname{Hom}_{\mathbb C}(\gamma,V)\bigr)
=\operatorname{Tot}\bigl((\gamma^*)^{\oplus(n+1)}\bigr).
$$
A <Hermitian metric> gives a complex-linear bundle isomorphism $\bar\gamma\cong\gamma^*$. With the Hermitian form linear in its first argument, it sends $\bar v$ to the functional $w\mapsto\langle w,v\rangle$. Linearity in the conjugate vector is exactly what makes this a complex, rather than merely real, isomorphism. Thus the bundle is $(\bar\eta_1)^{\oplus(n+1)}$, with the conjugation bars present in the original PDF.
Collapsing the closed subspace $\mathbb{CP}^n$ in <compact Hausdorff space> $\mathbb{CP}^{n+m+1}$ gives the <one-point compactification> of its open complement: neighborhoods of the collapsed point have compact complements, and conversely every compact subset of the complement is closed in the ambient space. The graph <homeomorphism> therefore extends over the added point. This proves the <projective-space quotient as a Thom space of conjugate tautological lines>:
$$
\boxed{\mathbb{CP}^{n+m+1}/\mathbb{CP}^{n}
\cong\operatorname{Th}\bigl((\bar\eta_1)^{\oplus(n+1)}
\longrightarrow\mathbb{CP}^{m}\bigr).}
$$
The graph coordinates also determine the normal <complex structures> requested in the second part. The copy $\mathbb P(W)$ corresponds to the <zero section> of the graph bundle, whose <normal bundle> is that bundle itself. For a coordinate embedding $\mathbb{CP}^k\subset\mathbb{CP}^n$, use $W=\mathbb C^{k+1}$ and its complementary coordinate space $V=\mathbb C^{n-k}$. Then the <normal bundle of a linear complex-projective embedding> is
$$
\boxed{\nu_k=\operatorname{Hom}_{\mathbb C}(\gamma_k,\mathbb C^{n-k})
\cong(\gamma_k^*)^{\oplus(n-k)}
\cong\bar\gamma_k^{\oplus(n-k)}.}
$$
The <complex structure> is the linear structure in these graph coordinates. Equivalently, applying the preceding <homeomorphism> with its parameters $n-k-1$ and $k$ identifies the <Thom space> of this <normal bundle> with $\mathbb{CP}^n/\mathbb{CP}^{n-k-1}$, after reordering coordinates. Define $\nu_m$ in the identical way. The Hermitian-metric identification with the conjugate line does not change the complex normal orientation.
The original coordinate embeddings may be nested, so first move one into transverse position. Keep $\mathbb{CP}^k=\mathbb P(A)$ for the first $k+1$ coordinates, and move $\mathbb{CP}^m$ to $\mathbb P(B)$ for the last $m+1$ coordinates. A unitary coordinate permutation accomplishes this. Every unitary matrix can be joined to the identity by diagonalizing it and varying its eigenvalue phases continuously, so this move is an <ambient isotopy>. Its trace gives a cobordism carrying the transported normal <complex structure>, and hence leaves the input cobordism class unchanged.
If $k+m\geq n$, then
$$
A+B=\mathbb C^{n+1},\qquad
\dim_{\mathbb C}(A\cap B)=k+m-n+1.
$$
Their projectivizations are transverse and meet in $\mathbb{CP}^{r}$, where $r=k+m-n$. The direct-sum normal structure from question 1 restricts to
$$
\nu_k|_{\mathbb{CP}^{r}}\oplus\nu_m|_{\mathbb{CP}^{r}}
\cong\bar\gamma_r^{\oplus((n-k)+(n-m))}
=\bar\gamma_r^{\oplus(n-r)}
\cong\nu_r.
$$
The quotient map defining the transverse normal-bundle isomorphism is complex-linear here, so it agrees with the standard normal <complex structure> and contributes no orientation sign. If $k+m<n$, the first and last coordinate subspaces have no nonzero vector in common, so their projectivizations are disjoint.
Consequently the <product of linear projective subspaces in complex cobordism> is
$$
\boxed{
[\mathbb{CP}^{k},\nu_k]*[\mathbb{CP}^{m},\nu_m]
=\begin{cases}
[\mathbb{CP}^{k+m-n},\nu_{k+m-n}],&k+m\geq n,\\
0,&k+m<n.
\end{cases}}
$$
The product degree is $2(2n-k-m)$, equal to the real codimension of the stated intersection when it exists. In the equality case $k+m=n$, the representative is one point with its complex-oriented normal space.
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