What order does Oracle's EXIST condition compare the sub-query results? I know EXIST works better than IN for larger sub-queries because it checks each sub-query record individually and stops once a comparison is found. 

Let's say I can order the sub-query in a way to make it more likely for the match to be at the top. Would EXIST actually stick to the order in which the sub-query is returned? 

I'm aware this approach is a bit of a double-edged sword, with specifying an order for a large sub-query having a negative impact on performance. I suppose I'm mostly just curious.
I would advise, don't try to out-think the optimizer like this. It doesn't always execute your query how you think it would be done based on how you wrote it. It may be able to take advantage of indexes or alternate join paths. And as mathguy said above, the difference between EXISTS and IN is much better than it used to be.

I would actually expect if you add an ORDER BY clause to your query, it might make it worse, because the entire result set might have to be retrieved, ordered, then checked against the EXISTS clause. But really, there's no way to know without an execution plan.

Just write your query so it retrieves the data you need, test it, then optimize it only if you need to. To quote Donald Knuth, "Premature optimization is the root of all evil."