But they are kids nonetheless, if you are in your "office" room (provided you have one) and next room your kids are screaming it's still disruptive (though headgear might help).
If you have >1 kids and they are home sick, I suggest just working from some other place for a couple days.
I do try and limit his time to about four hours a day, then I deactivate his WiFi and send him outside to play until mom gets home.
It really depends on the age, I'm struggling to get my child to poop in the toilet, I'm not sure he's ready to learn not to barge into the office when he knows his dad is in there.
This approach actually worked quite well for me. For the first month or so there were some hurt feelings, but once everyone adjusted to the new "normal", everyone was fine.
I've worked from home for four years now. We have a 6-year-old. And she has been taught not to bother daddy when the office door is closed.
If she comes up to bring me food, she is quiet coming in, and quiet going out.
As far as network interruptions, I have two Internet connections coming into the house, and my router fails over when the primary connection goes down.
Every single last issue you mentioned is solvable. If you're going to work from home long-term, then buy equipment that works for you.
In my case, I have a desk that raises and lowers, and my display is mounted on a VESA arm so that I can alternate between sitting and walking (I have a treadmill as well. You need to keep healthy!)