To add to the currently accepted answer:
Yes. The $$O\left(\frac{d}{\varepsilon}\log\frac{1}{\varepsilon}\right)$$ sample complexity upper bound holds for proper PAC learning as well (although it is important to note that it may not lead to a computationally(although it is important to note that it may not lead to a computationally efficient learning algorithm. Which is normal, since unless $\mathsf{NP}=\mathsf{RP}$ is it known that some classes are not efficiently proper PAC learnable. Cf. e.g. Theorem 1.3 in the Kearns—Vazirani book you mention). learning algorithm. Which is normal, since unless $\mathsf{NP}=\mathsf{RP}$ is it known that some classes are not efficiently proper PAC learnable. Cf. e.g. Theorem 1.3 in the Kearns—Vazirani book you mention, or [1]). This is actually shown in the Kearns—Vazirani book (Theorem 3.3), since $L$ there is a consistent hypothesis finder with hypothesis class $\mathcal{H}=\mathcal{C}$. See also [1].
Unknown. Hanneke's algorithm [2] is an improper learning algorithm. Whether this extra $\log(1/\varepsilon)$ factor in the sample complexity can be removed for proper PAC learning (information theoretically, i.e. setting aside any computational efficiency requirement) is still an open question. Cf. the open questions at the end of [3]:
Classically, it is still an open question whether the $\log(1/\varepsilon)$-factor in the upper bound of [1] for $(\varepsilon, \delta)$-proper PAC learning is necessary.
(Footnote 1 in the same paper is also relevant)
[1] A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth. Learnability and the Vapnik-Chervonenkis dimension. Journal of the ACM, 36(4):929–965, 1989.
[2] S. Hanneke. The optimal sample complexity of PAC learning. J. Mach. Learn. Res. 17, 1, 1319-1333, 2016.
[3] S. Arunachalam and R. de Wolf. Optimal quantum sample complexity of learning algorithms. In Proceedings of the 32nd Computational Complexity Conference (CCC), 2017.