IANA expert on smart plugs, but the typical space heater is very high wattage. Like near the max ampacity of an ordinary outlet. I’d be cautious about using a smart plug unless I knew it had the full 15A or 20A capacity.
All the space heaters I’ve used had thermostats. So leaving it plugged in / turned on 24/7/365 was just fine. It ran when it needed to and stopped when it didn’t.
No, you should only use a space heater when you are awake and at home to watch it; it not safe otherwise. I suggest adding more baseboard heaters instead.