Yes, this is exactly why the 16 dimensional sedenions are not in the same category as the others. For it to be a division algebra, you must be able to reverse any multiplication that isn't by zero. This is just the normal concept of division in the real numbers. It also allows us to define division in the complex, quaternion and octonion numbers. However, it is possible to multiply two non-zero sedenions together and get zero. This breaks this property completely.

> Is there a possibility of things that are distinct from our concept of numbers but are none-the-less useful being discovered in the future?

Absolutely! These four numbers systems are a tiny portion of spaces studied by mathematicians. There are a lot of interesting things that can be said just by generalising a few properties of numbers (e.g. without requiring that we can divide, or without requiring that we have both addition and multiplication). There are groups, fields, rings, algebras, topological spaces, and all sorts of interesting things that aren't 'numbers' as such.