Checkout instructions provided by Ledger Nano S support at
https://support.ledgerwallet.com/hc/en-us/articles/115005530425-How-to-install-and-use-Neo-NEO-.
The url you provide leads to a place where the wallet looks just like the one that you should be using. Nevertheless I would not trust any site other that that referenced in Ledger’s instructions just to be on the safe side and to avoid fake wallets.
Yes, you will earn Gas on ledger for Neo. What you will need to do is to claim it every now and then from the Neo wallet interface and sign the tx twice from the ledger nano s during the process. Just click on the Claim Gas button from the wallet and follow instructions.
