I think most locks will offer option number 1.
For 2: The Schlage locks state this
Built-in alarm technology senses potential security breaches at the lock, while the low battery indicator offers advanced warning for battery replacement
Also I found August has the following alerts:
The Door is locked or unlocked manually
This alert notifies you when someone operates the August Smart Lock manually by hand or using a key.
The door is locked or unlocked by a specific user
This alert notifies you when a specified guest or owner operates the lock via Bluetooth. This includes unlocking the lock using the app
The door is left ajar (3rd Gen Smart Lock and Smart Lock Pro only)
This alert notifies you if the door is left open for a specific amount of time. Specify the amount of time in the lock settings.
The door is Auto-Locked
This alert notifies you when your August Smart Lock Auto-Locks. Auto-Lock can be set up in the lock settings page.
The door is One-Touch Keypad locked (only if August Smart Keypad
paired with lock)
This alert notifies you when the August Smart Lock is locked using the One-Touch Lock feature on the August Smart Keypad.
For 3: If the door cannot get on Wifi, it cannot contact you. The only way I could see something like this working is to have a device constantly pinging the door lock. If it can't find after a certain interval, send an alert.