Yes, Moto Z2 Play does support fast charging. It takes just 30 minutes to charge 50% battery. You can get about 8 hours of use by charging for just 15 minutes.
No. Moto Z2 Play does not have HDMI port but you can use HDMI connector.
Yes, it does support OTG.
Yes, it does support 4k Videos shooting as well playback.
Yes, Moto Z2 Play comes with latest Android 7.1 Nougat OS and does support Screen Cast.