1) To my understanding, this is not an online wallet: the website simply generate a wallet address, and gives me the public and private keys, which I can then safely store offline or on a piece of paper, if I wish so, right?

Yes, you are all right, MEW is an online wallet, which can be accessed easily anywhere and anytime, can be via pc or laptop device, can also be used on android connected internet network, you just need to save its private key only, you can access it easily