I would like to ask since a woman can recite Qur'an while in her menses:
- Can she read it on her phone, like touching the verses in her phone?
- Even in her menses does she have to perform wudu before reciting or doing dhikr?
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As a general Islamic rule, it would be considered as a haram practice to touch the script of the Holy Qur'an with any part of one's body, without performing Wudhu. But in regards to your second inquiry that you asked
1) Can she read it on her phone,like touching the verses in her phone?
The response is that touching the Holy Qur'an through cellphones is not considered as a haram practice. Since it would be touched indirectly. In truth, that is the light/glass of the mobile phone that you touch, not the real Quran.
In regards to your second inquiry that you asked
“2) Even in her menses does she have to perform wudu before reciting or doing dhikr?”
You should pay attention that it is not obligatory (Wajib) to perform wudu either. Since the haram practice is touching of Quran directly.
Note: As a significant point, when some women is in merses, she should not recite the verses which are Wajib-al-Sajdah (obligatory to Sajdah), even it some Marja'al taqlid declares that she ought no to recite all those Surahs (which have Wajib-al-Sajdah verses). Otherwise you can recite the rest of the holy Quran (except those 4 Surahs according to Shia's perspective)