Shokugeki rules elicit that the conditions of winning and losing should be set beforehand. Souma Passes if he wins, gets expelled if he loses. Hayama gets nothing if he wins/loses. Though he had a deal with Azami to be expelled.
So, No. Souma is NOT a member of Elite 10.
In fact in the next chapter he discusses that if he, Megumi and Aldini, grab the Elite 10 seats. They have a chance of reversing Center's expulsion.
As discussed in the related question,
Each side must lay out their "compensation", or something they must give up if they were to lose the challenge. Both sides must agree that the compensation of the duel equate with each other.
Source: Wiki: Shokugeki
This is established pretty early when Souma challenges Isshiki for his 7th seat earlier!