Yes, that's fuse wire.
It's ceramic, I think. I've never considered it to be asbestos.
Yes I've got done 15's and 30's in there too - as well as the wiren one.
The one in your second photo is rewirable, it does not take a cartridge fuse. And given the multiple strands of wire visible it doesn't look as if it currently contains fusewire - if some idiot has put a core from a piece of flex in there it is very dangerous.
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