If I use a company iPhone on my own home Wi-Fi, does the employer still have visibility on activity? If so, how much assuming there are no hidden keylogging programs.
If the device is enrolled on your company's network (using a device enrollment / Mobile Device Management process), they could have a lot of visibility. iOS allows for a "always-on VPN" (link). This allows for either certain or all applications to use a particular VPN connection; which could be your place of employment. With the implementation of enforced / transparent proxies and custom CA's, he/she can also intercept SSL data easily and see what you do on the network.
So in essence, your employer could see a lot of traffic going back and forward to your device, manage your apps, remove all/some of your content, etc.