I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in C^2 for the standard symplectic structure, mentioning that this were the only compact surface for which the problem were open. Question 1: Where to find references for the other surfaces? Question 2: What is the current status for Kleinian bottles? Do there exist written proofs now?

Just googling "Lagrangian klein bottles" returns the following : Stefan Nemirovski, Lagrangian Klein Bottles in R2n Geometric And Functional Analysis Volume 19, Number 3 (2009), 902909 the freely accessible arxiv version is at http://arxiv.org/abs/0712.1760 abstract : "It is shown that the ndimensional Klein bottle admits a Lagrangian embedding into R^{2n} if and only if n is odd." There was a previous result by the same author about the $n=2$ case, but it had a flaw initially, and was completely rewritten. See http://arxiv.org/abs/math/0106122 There is also another proof of this result by V. Schevchishin http://front.math.ucdavis.edu/0707.2085 


While working on my thesis, I (by accident) constructed a Lagrangian Klein bottle in $(S^2\times S^2, \omega\oplus\omega)$. The construction works in $S^2\times D^2$ as long as the area of $D^2$ is strictly greater than half the area of $S^2$. The method is to pick a Hamiltonian on $S^2$ that rotates $S^2$ halfway around, and realize it as the monodromy around $\partial D^2$, of a symplectic fibration over $D^2$ with fiber $S^2$. There is a circle that goes through the two fixed points of the rotation, which then traces out the Klein bottle. See McDuff's ``Geometric Variants of the Hofer Norm'' for the precise version of how to realize such a monodromy and the constraints: the area of the base is an easy way to think of it, but it's really a ratio of volume to fiber area. As I understand it the paper by Nemirovskii called ``homology class of a Lagrangian Klein bottle'', is flawed, because it predates the accepted proofs of nonexistence as described by others. I would love to know if its main claim is true, that a Lagrangian Klein bottle in a symplectic 4manifold must realize a nontrivial second homology class with mod 2 coefficients. This is true for the example above. (Might need to ask this as a question.) 


May be this is relevant: http://arxiv.org/abs/0712.1760 Lagrangian Klein bottles in R^{2n} Stefan Nemirovski It is shown that the ndimensional Klein bottle admits a Lagrangian embedding into R^{2n} if and only if n is odd. Comments: V.2  explicit formula for the Luttingertype surgery; 6 pages 


Just to explicitly answer the first part of your question, the original version of Nemirovski's first paper (http://arxiv.org/abs/math/0106122v1) surveys what is known about the other surfaces. Namely:
That leaves k=0 for Nemirovski/Schevchishin. 

