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
or Auto-Unlock.
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.