ECDHE without initial handshake

Is it possible to do a key agreement with ECDH ephemeral-ephemeral without requiring an initial handshake first?

Yes, however this is vulnerable to man-in-the-middle (MITM) attacks.

Is it possible to have the key agreement in the same message as the encrypted data with ECDHE?

For the receiver to use an ephemeral key, they would have to send it before you can use it for the exchange. Alternatively they could load a number of ephemeral keys onto a trusted third party server to be retrieved by senders later (this is kind-of what is done by Signal). But usually you'd assume the sender just uses the recipients static key so they know it is going to the right person. In this case, all the ephemeral randomness is provided by the sender, so you really have a KEM (key encapsulation mechanism) rather than a key exchange.

So essentially, if you're willing to forgo authentication and risk impersonation / man in the middle attacks, you can achieve perfect forward secrecy with a KEM, and you can have the receiver regularly rotate the receiving key so they don't have to worry about secure long-term storage (a semi-static key).