1: yes 125khz
2: EM mode is the normal mode. The other mode is specifically for cloning an HID brand prox card to it. Quirks of the T5577 & cloning tags to the xEM
3: The PN532 is a 13 mhz reader, the xEM is a 125khz tag, They are not compatible. You can use the XEM access controller https://dangerousthings.com/shop/xem-access-controller/ though this is meant for direct control of whatever you want to control with your tag, not connect to an arduino. Or you can use something like the ID-12LA module sparkfun has. They also have a starter kit. https://www.sparkfun.com/products/13198 This spits out the ID of the tag which you can connect to an arduino and read in.
You have the wrong tag for phone unlock. You need the xNT which is NFC compatible… Your phone will not read the xEM. Read up here. X-Series Implantable Transponder FAQ
Also ,while it can be done in certain ways with third party apps, the native NFC unlock function has recently been removed from Android.