In Dutch culture it is far more common to keep shoes on literally everywhere, so perhaps I'm biased, but I would think that in an emergency where every second counts, these people can be forgiven for not adhering to the strictest etiquette?
I'm not surprised the person last mentioned in the article, who suggests exactly that, got 800 likes on their comment. I am surprised that it even needed to be said.
Now I'm suddenly wondering (probably stupidly) whether firefighters also take their shoes off when entering a burning building.
So, I can’t honestly personally understand taking your shoes off in an emergency, however I am sure they have gotten very efficient at it.. I can say that if they didn’t it would cause damage to the home, many Japanese homes, especially traditional but also in general just aren’t built to take shoes. Tatami mats, and even hinoki floors despite being “hard” for a softwood, is still a softwood, so some level of genuine damage would occur. Again of-course versus the emergency in saving a human life, I’m not quite sure I can understand it.
I think we can roughly split paramedic visits into “patient needs to get out ASAP” and patient can’t get out alone or with help of friends and family”, so I would hope they keep their shoes on for the first type.
But ambulances in with full sirens begging over their speaker-system to have pedestrians at Shibuya crossing let the vehicle through, is probably the worst thing about Japan.
As for your second question, I’m reminded of Bob Mortimer in Would I lie To You: “… then the fire brigade came and ruined the whole house” - “Bob, surely the fire was ruining the house” - “No, water damage”. Maybe if those darn firefighters had taken their boots off, there had been a bit more left of Bobs childhood home.