I don’t know Inkscape, but I can explain what I see in Illustrator and I assume the same concepts apply.
If I just naively select the whole logo (apart from the text and emblem) and set it to filled, this happens:
Since there’s a lot going on there, let’s focus on one thing. First of all, it’s one big compound path, so start by releasing that and just looking at the upper right where the sword hilt is. Again, it fills strangely:
But now it’s a bit more straightforward when you look at one of the paths in isolation:
It’s not closed, and it doesn’t really represent a whole shape. So if you connect the start to the end and fill it in, you get that part of the weirdness above. Actually, if you stare at that thing closely and trace where the lines go, it has a sort of M. C. Escher thing going on. What’s inside and what’s outside is not exactly clear.
As for how to fix it, well, I’m a technician and not an artist. I imagine the best way is to manually clean up the paths and redraw the nasty bits. 10 seconds with the Shape Builder tool in Illustrator can repair the paths in place, but it still requires a bit of touching up and I daresay it’s not what you’d even want.
Hope that helps explain things a bit. It’s a really long winded way of saying the paths are all screwy.