Is a woman with these obsessive doubts considered a Muslim, and is it permissible for her to perform the Hajj pilgrimage while suffering from them?

The questioner's denial of her whispers (waswas), her fear of them, and her supplication to Allah for their removal, is evidence of her faith. It has been narrated in the Hadith (definition): "That is pure faith," meaning that deeming such thoughts grievous is pure faith. We advise the questioner to strive diligently in repelling these whispers, to return to observing prayers, and to go for Hajj, for indeed, the plot of Satan is weak.