Yape. Even you manage to flash everything in local mode. U still need to leave your Router On. with/without internet connection.

Depends on what you want to achieve.
You can set a schedule on the app and then delete the app.
The schedule will on or off the devices without your interaction.
Yes, must have internet connection if the devices depend on the cloud.

if your whole house using wifi based bulbs and switches then mati loh....hahaha
just last week, i woke up at around 1.30am because my bedroom lights were still on. My wifi bulbs didnt turn off on their own as per scheduled.
i use phone to off also cannot. I thought my Internet or Wifi down, but I was able to surf Lowyat forum.
then i check my LIFX bulb's server down or not but cannot enter their website.
Long story short, finally i found out that my wifi was down. I cant even enter my router's setting to reboot. Strangely I can still surf lowyat forum.
Niamah, had to get out from bed and Off On the router.

The next day, I tried looking for a device that can reboot my router when it detects the Internet is down.

I think this is not something specific.

The smart switch with no N has maximum total load. So, if total load cannot exceed this max. Maybe this is the reason. If the switch require N, no such restriction.

Fake news. Been using Sonoff T1 3 gang with fan for nearly two years with no issues.

No you cannot, at least not natively
If you are already using Xiaomi, why not just use Xiaomi to do all that?

I was just joking but yes need that sensor, thus there is not need for the trigger when opening door then. It becomes redundant.

Opening door trigger, the system need to know you are entering or leaving the house 

Can, under Home Assistant provided your home alarm, door sensor and light and fan switches can connect to it.

It’s two different system so if you want connect them together you will need another system to bridge them
What I meant is if you are using Xiaomi door sensor to trigger, why not use Xiaomi to switch on the lights as well?