x is private by virtue of being the conveyed by an ECDH key negotiation. No external communication is required.
Do the wallets still need to connect directly, without touching the blockchain? How the receiver would learn the amount then?
No, the idea is that addresses in Elements alpha contain an ECDH pubkey, so the sender can (non-interactively) send secret information to the receiver using ECDH key exchange. The amount of the transaction output is embedded into the range proof, without taking up more space (it's actually XORed in). No prior interaction is needed between sender and receiver (that's a pretty fundamental requirement of course).
Do I have to be online at the same time as you?
No, see above.
They're required for outputs only, and technically only when there are multiple outputs. Inputs are already known to be in range by virtue of having been created as outputs.
This is about range proofs but I asked about commitments.
Both Pedersen commitments and range proofs are published to the network for each vout (utxo).
You might find this doc I wrote useful if you're looking into the details:
https://github.com/AdamISZ/ConfidentialTransactionsDoc/blob/master/essayonCT.pdf. For example, the diagram at the end (page 20) of the transaction layout.