To summarize it for you,
IF you are just an indie/entry level dev., use Corona SDK and Corona SDK is entirely free. Enterprise lets you call APIs you have created yourself for doing specific things with your app. Corona uses Lua, if you have your own library in C++/Java and you need to use those, you must have Corona Enterprise and that is not free.
For your specific concern, yes Apple game center can be used in Corona SDK and is entirely free. In fact, you can also do In App Purchases and ads (iAds, AdMob and Vungle) through Corona SDK and they are free too( Corona SDK is free and as of yet no revenue sharing/splitting). FYI, if you wanna publish your apps in the stores, Apple has a $99 yearly developer fee and Android has a $25 registration fee Those are irrespective of what platform (in this case Corona SDK) you chose to use, and you pay that to Apple/Google. Corona itself is free, unless you wanna use your own Java/C++ library. Hope that helps.
Here are all the functionality you get with SDK:
https://docs.coronalabs.com/api/index.html Go to gamenetworks to see Apple Game Center.