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It will be perfect, you just need to set it up right.
You keep your router (with its internal network switch). This makes you an internal LAN, separate from the WAN/internet. You use the AP, plugged into the router, to set up a WLAN that allows your to talk to the LAN over WiFi (with sweet speeds). Turn off the WiFi on your Zyxel to avoid conflict.
If you want to completely open one computer to the world (are you really sure?) then you need to set up the router to do this for you - it's the gatehouse between the external WAN and internal LAN. You can either put that computer into the DMZ (yes that means de-militarized zone, it's a gaming feature most routers have) and it will appear completely open - this is dangerous - or you can selectively forward ports on the router's firewall to your internal device. I would suggest the latter.
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You can do that.
Switch off the built in wifi on your router
Plug the Unifi AP into the router and use it as your wifi access point throughout the house.
Anything else that needs a wired connection to the internet can still be plugged directly into the remaining ethernet ports on the router.
Your current broadband router (with shit wifi) will continue to work as a router, you just won't use it for wifi as you'll have the Unifi AP plugged into it.
Edit: And set up port forwarding on your existing router.
Port forwarding is a different thing to do with firewalls.
The UniFi APs will provide a standalone WiFi network from an ethernet source. You can set the SSID of the AP to be the same as that of an existing network (along with all the other security settings) and you will have some amount of range extension. Alternatively, you can set up a parallel WLAN and choose the stronger of the two.