She may not be eligible for a permit.
In order to be eligible for a permit, a person must not be holding citizenship other than Chinese (Mainland/Hong Kong/Macau). Otherwise, that person must use the passport with foreign citizenship to enter Taiwan.
For example, a Hong Kong Permanent Resident with British Citizenship must use British Citizen Passport to enter Taiwan as visa-free travel.
Since your mother is born overseas, she may have a foreign citizenship. If this is the case, she will not be eligible for the permit.
Addition:
In answering your questions:
1. See here:
http://www.teco-hk.org/ct.asp?xItem=111598&CtNode=7511&mp=10
Also, the proof of renouncing the previous citizenship will be needed as well.
2. Within 2 weeks.
3. Yes. There are many reasons. But Taiwan is lenient towards HKSAR and Macau SAR residents. So unless the applicant has broken any Taiwanese law, the application will be approved.