1. I'm not aware of any other wallet apps that support Bitcore. So if you want to claim BTX or register for the airdrop, I guess you'll need to use the Bitcore-QT wallet for now.
2. If you scrolled a bit further down the homepage for "Bitcore" (The actual website is
https://bitcore.cc/ NOT bitcoincore.cc)... you would find the links for downloading the wallet.