Many highly rated video doorbells don't require a subscription, such as the Reolink video doorbell cameras. They store the video locally, usually on a microSD card or in hard drive. This way you don't have to pay a monthly fee and you have more control over your recordings.

It depends. Some doorbell cameras do not record continuously. Instead, they only start recording when they are triggered by motion or an event. Once triggered, they send a notification to your phone, ensuring you do not miss the recorded clip.
However, the Reolink video doorbell PoE/WiFi supports 24/7 recording. You can set Reolink Video Doorbell PoE/WiFi to record 24/7 continuously using a micro SD card (not included) or an NVR.

It depends. Most wireless doorbell cameras typically require a Wi-Fi connection to function effectively. Wi-Fi allows the doorbell camera to connect to your home network and transmit data, such as video footage and alerts, to your smartphone or other devices.
However, PoE doorbell cameras do not necessarily need Wi-Fi for their primary functions. POE cameras receive both power and data through an Ethernet cable connected to a POE switch or injector.
If you use Reolink doorbell cameras without an Internet connection, the doorbell camera can still detect motion/person and record videos to the SD card. However, it cannot send alerts to your mobile devices, and you cannot live view it.
If the chime is connected to the doorbell, it will chime when the doorbell is pressed, regardless of the Internet connection.

You can install the doorbell camera at entry points of your property, such as the front door. It is recommended to mount the camera 1.2 meters (approximately 4 feet) above the ground. A 15° wedge is included to help you set up or adjust the camera to the optimal angle.