I had not heard of Wheel theory, quite fascinating.

But eek! From the wikipedia article:

    0x ≠ 0 in the general case
    x − x ≠ 0 in the general case

Like, you get the ability to divide by zero, but at what cost? Multiplying by zero is also not zero! Seems bonkers.

Are there any practical uses for this kind of algebra?

Likely no (the peculiar distributive laws seem unusable to me). But I have come across similar rules that I think may have a real usage. It sometimes seems to me that it might make more sense to keep 'factors' attached to 0s, such that x-x = (1-1)x = 0x might be 'sound', and to keep track of 'powers' of 0s by having factors of 0 attached to 0s: (0)0 = 0^2 != 0, etc.

If you keep factors like this, then you could implement L'Hopital's rule without the result only being true if considered under a limit: say, lim(x->0) (5x^2/ 3x^2) = 5/3 could be computed as (5)0^2 / (3)0^2 = 5/3.

This is not like Wheels though; it requires that any power of 0 be a distinct number. I of course have no idea if it is sound or meaningful, but I do find myself thinking about it a lot.

Those "factor" zeros look like infinitesimal forms, which are well studied.

There is a similarity, but no one claims infinitesimals are _literally 0_, and certainly no one would claim that 2-2 = 2 epsilon.