I proceed assuming you wanted addition to stay the same (since you said nothing about changing it.)
If you don't care about distributivity, or you don't care about addition period, then yeah, you can take whatever function you want $\mathbb R\times\mathbb R\to \mathbb R$ and sometimes the order of the inputs will matter.
If you do care about addition, then you can propose whatever weird rules you want for a binary operation, but it will often be disastrous for other properties that we value about multiplication, like distributivity.
Take the second proposed axiom for example: $n\otimes(-m)=nm$
If we wanted distributivity, $n\otimes m + n\otimes(-m)=n\otimes(m-m)=0$, so that $n\otimes(-m)=-(n\otimes m)=-nm$. With your axiom above, we'd have $nm=-nm$ so that $2nm=0$. But this is using regular multiplication and we know that's not true in the real numbers for nonzero $n,m$.
I think there are some oddball binary operations on $\mathbb R$ that can be useful, but by and large the most-used ones are those which cooperate with addition, so that you have a ring structure.
How could this math be applied?
Try not to fall down the rabbit hole of spending time with "solutions looking for problems" and try to get into the mindset of "problems looking for solutions." Almost always (or, always?) the most fruitful mathematics are generated in the service of solving a problem, not the other way around.
BUT perhaps you meant to ask something more like this, which I think is a fair question:
What are some examples of noncommutative binary operations on the reals that have applications?
Well, now that I think about it, two come to mind:
$a\otimes b=a/b$ and $a\otimes b=a-b$. These 'have applications' but their study does not seem to go very far beyond what we already learn with regular multiplication.