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 


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. 

