Proof of NP Completeness of set-partition problem
No, your can't arbitrarily choose $t$ so your proof is incorrect.

In fact, you have build a reduction from the following variant of subset sum problem to partition problem:


  Given a set $S$ of integers whose sum is $s$, is there a subset of $S$ whose sum is $s/2$?


This variant is different from the normal subset problem, and you cannot assert it is NP-complete (though it in fact is), so your proof is incorrect.

You can find a complete proof here.