In brief, is it permissible for my husband to marry a second wife secretly while neglecting his first home, and is it permissible for me to ask him to divorce her so that stability may return?

The servant is happiest when content with Allah's decree and law. Had you been content with what Allah allotted you and treated your husband as Allah commanded, matters would have been better. His second marriage is valid if its conditions and pillars are met. However, if he married secretly without a guardian or two witnesses, the marriage is void. It is not permissible to demand the divorce of the second wife, bearing in mind that marriage can be obligatory or forbidden depending on the circumstances.