There's a good discussion thread about
hotlinking over here. I think particularly within neocities its fine because its all coming from the same host regardless of what site the badge is on.
As for badge etiquette! I have OPINIONS! Firstly, no you don't need to ask to display someones badge, if someone has put a badge on their page saying "this is my badge", that means they put it there intentionally so you can take it and use it. Some people will very much like it if you tell them you've done included their badge on your site, because they might be interested in your site, and they might want to link back too. However that's another important point; no one owes anyone a link-back, just because someone put your badge on their site does not mean you have to do the same. Putting a badge on your site is simply a way of saying "i enjoyed this other site, here is a link to it".
However there's also something quite personal about a badge as Adam says; they can represent a lot to someone, and I think its really important to be respectful about that. If you are going to put someones badge on your site, be respectful to how you do it. Don't just put it there to fit your layout, and don't hide it in a scrolling marquee that visitors will never see. Badges are not decoration, they are part of the person who created them, and putting them on your site is like inviting part of that person into your home, you have to treat them well.
The same goes for badges you design. When you design a badge, you're creating something that someone else will bring into their virtual home, its a gift to them; so you should try and make it a good gift. One that represents you, but also doesn't try too hard to grab attention, and tries to add something good to whatever site its put on