Is there an analog of the statement of "every 2d oriented surface is a complex manifold"?

I saw a theorem in Blair's book, that "every 3d contact metric manifold is a strongly pseudo convex CR manifold". And we also know from Lutz and Martinet that every compact 3d manifold has a contact structure (and therefore a contact metric structure), it seems the analog would be "every compact 3-dimensional manifold is a strongly pseudo convex CR manifold".

So my question is:

1) Is this the right analog?

2) Introductory/review references on the 3d analog of "pseudo-holomorphic curve in symplectic manifold" (for instance, references on how they are defined, basic properties, counting invariants analog to G-W invariants)?