When can L'Hospital rule be used on Multivariable limits.

Bear in mind the L'Hospital's rule goes for single-variable limits, only. Checking a lot of different paths will not guarantee the existence of the limit. But if you find any two different paths which give you different numbers, then the limit does not exists.

That being said, once you have chosen a path, the limit becomes a single-variable on, so yes, you can use L'Hospital. For example:

in trying different paths, say $f(x,x^2)=\frac{x^2(1−x^3)}{(1−x)}$ would it now be valid to use L'hospital? because the $y$ is gone and we would have $0/0$ as $x →1$?

Here you chose a path, and now you have a single-variable limit. You can use L'Hospital.

Edit: It seems that there is a sort of L'Hospital's rule for multi-variable limits, as pointed by Git Gud in the comments. Check it out.