If I hosted multiple piece of ICE in awakening center, can I rez another hosted piece of ICE after the runner encountered the previous hosted ICE that I forced him to do so using it's ability?

I'm gonna say no. The condition "whenever a runner passes all of the ice protecting this server" is fulfilled only once. "Ice protecting a server" has a specific meaning: ice installed in front of it. The ice on Awakening Center is installed on a card in the server and therefore does not qualify as "ice protecting this server", so the condition for Awakening Center's ability is not triggered again when the runner has finished the encounter with the Awakening Center ice. And I take the "rez a piece of ice on Awakening Center" to mean just a single piece of ice.

the text says 'a piece of ice'. As in singular.

You may only rez one.