An answer to question 2: Shelah constructed a model with exactly one $P$-point (up to isomorphism). In fact, the one $P$-point is a selective ultrafilter. You can find the construction in section XVIII.4 of Proper and Improper Forcing.

An answer to question 2: Shelah constructed a model with exactly one $P$-point (up to isomorphism). In fact, the one $P$-point is a selective ultrafilter. You can find the construction in section XVIII.4 of Proper and Improper Forcing.

An answer to question 5: If there are exactly $2$ classes of near coherence, then $\mathfrak{u} < \mathfrak{d}$. (You say this in your post, just before Question 4.) In other words, there is a non-principal ultrafilter generated by fewer than $\mathfrak{d}$ sets. Ketonen proved that any ultrafilter generated by fewer than $\mathfrak{d}$ sets is a (non-selective) $P$-point. See

J. Ketonen, "On the existence of $P$-points in the Stone-Cech compactification of the integers," Fundamenta Mathematicae 92 (1976), pp. 91-94. (link)