The extension cord "alternate electricity system" as they call it seems particularly harrowing. You had better make sure that you know what the load rating is on each of those cords and what they are going to be supplying (or better yet, just don't do this). Unlike household wiring, there will be no breakers.
The "load rating" on the cords exceeds the total power delivery capability of the system. It has its own circuit breakers.
The manufacturers are well aware of risks, for liability reasons if nothing else, and have contained them in the uniform-electrical-code mandated manner. The system is no more risky than any other plugged-in appliance. We don't need to invent problems. Extension cords are just extension cords.
no they're not. Not when you're dealing with potentially large loads. an extension cord rated for 13 amps with 16 gauge wiring, with a 110v dryer running on it, will cause a fire, sooner or later. A proper 10-gauge, 20 amp-rated extension cord will not.
that this system won't exceed ratings is a fair point, but extension cords are not just extension cords.
Obviously the extension cord you use needs to be rated for the power it carries. Anyone buying extension cords can see there are different sizes. They are well-labeled, and it is easy to look up the gauge needed for its length and what is to be plugged into it. A correctly-rated extension cord is no more hazardous than the same-rated Romex, and often cheaper.
All the appliances mentioned are rated to be used with appropriate extension cords, and cords are sold at retail for such use.
"obviously" is doing a lot of work there. What's obvious to you isn't obvious to the next guy, who's just going to look for the cheapest cord there is.
They're not well marked at Home Depot, you have to bother to look for the right gauge, the cables aren't even necessarily in the right place, and if you don't know jack about being an electrician, it's easy to buy a cheap 2-prong 18-gauge indoor-rated cable for use outside in a wildly inappropriate fashion. No one at Home Depot is going to stop you and ask what you're about to do with the cable.
yes, For people with half of a clue, it's possible to buy the right cable and be safe, but we have to plan for the lowest common denominator, and tell people that extension cords are just extension cords when they come in all sorts of flavors, some of which are happy to burn your house down, is irresponsible.
The store brand, which is general most prominent, gauge is color coded: 6AWG - Gray / 12AWG - Yellow / 14AWG - Pink / 16AWG - Orange. There's one exception to this where 14 AWG cable is using orange labeling normally used on 16AWG wire but that's the opposite of a safety issue.
And they have generally have the rated amperage for the length of the cable.
This all falls under "it is possible to kill yourself by stupidity", with nothing specific to powering appliances from a battery/inverter. People who take the initiative to set up such a system self-select for attention to detail. It is a lot easier to use the wrong cord with no battery system than with.
The extension cords mentioned are all interior-to-interior. There is no magickal difference between an extension-cord wire and a like-gauge Romex wire. The former bends easier.
I've run my share of long extension cords and chained power strips, running every which way, in both rentals and owned dwellings, powering space heaters and everything else. One thing you really need to watch out for is the connections - even a good tight plug+receptacle has higher resistances (electrical+thermal) than the cable itself or a hardwired terminal screw clamp. An older looser-fitting receptacle can make poor contact with higher resistance and even arcing, making a lot of heat. And also the risk of physical damage to the cable of an extension cord is generally higher. So do what you will, but make sure you're inspecting such runs often, especially for excessive temperature rise at the ends.
