Je is pronounced at the end of the syllable containing it. The ending e of je is generally silent but not in verses like N’ai-je donc tant vécu que pour cette infamie.

Essentially in written French. Sometimes also in formal spoken French, puis-je vous emprunter votre stylo ?

Note that veux-je is quite rare and hurting many French ears. Some authors even use the barbarism voulé-je instead to avoid it…

Yes, the pronunciation is the same [ɛ]. The accent has precisely been changed to match it.