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You mention an interest in symplectic geometry and Fukaya categories. A very natural problem in the field is given, say, a Liouville domain (a very reasonable class of exact symplectic manifolds) what ...

In the mathematics literature, this example is a basic one for SYZ mirror symmetry, going back at least to Gross' seminal paper
http://arxiv.org/pdf/math/0012002v2.pdf
See Chan,Leung,Lau JDG 2012 ...

I'm not sure I understand the question, we have a graded commutative algebra --- how can you deform it to an $L(\infty)$ algebra? However, one can simply regard poly-vector fields as a Lie algebra and ...

Your $L_\infty$ algebra is really only a good notion up to quasi-iso, so your enveloping algebra makes most sense as a dga defined up to quasi-isomorphism. From this point of view, this is due to ...

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