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Why are symplectic toric varieties projective?

Not a full answer, but a partial answer to your question 1a. Torus-invariant Kaehler metrics (in particular complex structures) were constructed by Guillemin, just starting with data on the moment ...
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Does there exist a point $ x $ of an affine toric variety $ U_{\sigma} $ such that $ x $ is not compatibly split?

Actually, on an affine variety, if it is $F$-split, then it is compatibly $F$-split with every point (possibly changing the splitting). This is not true in the projective case (an ordinary projective ...
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