The best question I can ask at this point is this: is there a way to order things or arrange things of which it is not even possible to use numbers or any form of numerical counting?
I think you will really need to be more clear. I have no clue what you mean. Every collection of objects (let's keep it finite) can be counted and thus can be assigned a number.
It is possible to order "un-countable" sets, if that is what you are talking about. For example, the set of all real numbers between 0 and 1 is uncountable and is a subset of the set of all real numbers between 0 and 2 which is a subset of all real numbers between 0 and 3, etc. We can "order by inclusion"- A comes before B if and only if A is a subset of B. Of course, that collection of sets is then countable. But "ordering" is in fact equivalent to "counting". If a collection of objects can be "well ordered" (given any two objects, A and B, we can determine whether A is before B or B is before A and each object has a unique "next" object) then the collection is "countable".
Right. Practically it seems that in this universe there is probably no way to do analysis without some form of numerical ordering. The only reason I brought it up was that I imagined a potential non numerical analysis would be the end result as the limit of the level of abstraction approached infinity.