I have tried to look for this Teshuva from Rav Elyashiv ZT”L and can not find it in vol 1-4, if you can please write where it is.
Regarding custody I looked it up and in Simin 82 Sif 7 it says that the mother always gets full and complete custody of both the boy and girl forever (when the boy turns 6 the father can say that he does not want to pay child support because he wants his son to teach him Torah, but this only applies to money not custody). This is all black on white ??”?, maybe you have a different edition.
A lady can force divorce in many cases for example (abuse, she can swear not to eat certain foods or be with him and many more types of Nedarim all brought in SH”A which come from Mishnayos). Also see IG”M simin 7980 where he writes that if a husband lost his mind a get might not even be needed. It was also stated in the Teshuva you quoted from Rav Elyashiv (where?) that if she gives a reason Beis Din should force it. The Gemara and SH”A also pasken that a lady does not have to live with a snake. In the times of Chazal or even the Rabeinu Asher these were enforced by beatings.