Is it possible for people from other instances to use those famous hexbear emojis?

Yeah, each one is an image link. For example, is

https://www.hexbear.net/pictrs/image/16a72a38-0fbc-45f3-bf59-dd9e6509f73e.png

So you can go to the image, copy and paste it into your comment, and use it from your own instance. You won't have the emoji picker unless you have a hexbear account, but you can bookmark and hold on to your favorites.