1. dom(≤) = cod(≤). That is, there exists U⊆S∪T such that, to use Haskell's notation, ≤ :: U -> U -> Bool.
2. You're defining ordinal numbers in the manner that Peano did. That is, O := { o∈U : o < O}.

Now, your question basically boils down to asking whether every set has a maximal element. Some sets do, but in general the answer is no. For instance, the natural numbers do not. Some supersets of the natural numbers do; analysts construct ∞ for exactly that purpose. I'm guessing that the set you had in mind is something along the lines of the surreal numbers. They do not have a maximal element, but once again you could easily construct one.