Can the backdoor in Dual_EC_DRBG be used to create a public key stream cipher?
You can do something like what you are suggesting. But, the EC_Dual_DBRG also has biases in the stream and so you cannot use it without changes (e.g., truncating much more). However, this is based on the same operations as ElGamal.

The public key is set up exactly as proposed. Then, to encrypt a message $m$ of any length, do:


Choose a random $r\in\mathbb{Z}_q$, where $q$ is the order of the Elliptic curve group
Compute $U=r\cdot P$
Compute $k = H(r\cdot Q)$
Encrypt $m$ with $k$ using any symmetric encryption scheme you wish