Monochromatic point sets in two-colored plane

I think the best starting point are surveys by R. L. Graham; for example:

Euclidean Ramsey Theory, Handbook of discrete and computational geometry, 153–166, CRC Press Ser. Discrete Math. Appl., CRC, Boca Raton, FL, 1997. (http://dl.acm.org/citation.cfm?id=285879)

Recent trends in Euclidean Ramsey theory, Discrete Mathematics Volume 136, Issues 1–3, 31 December 1994, Pages 119-127, https://doi.org/10.1016/0012-365X(94)00110-5

In particular, if you allow scaled copies, then every finite configuration $P$ has property $M$ (a theorem by Gallai and Witt). This should answer all the questions 1–4 for finite configurations $P$.

Also see this recent paper by J. F. Alm for a strenthening of Gallai's theorem (he shows that $2^{\aleph_0}$ monochromatic homothetic copies of $P$ exist):

https://arxiv.org/abs/1304.3154,

http://intlpress.com/site/pub/pages/journals/items/joc/content/vols/0005/0004/a003/index.html

Not an answer, just an illustration: Permit me to mention Ron Graham's challenge from 2003:

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    ^{(Figure from Computational Geometry Column 46.)}

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The citations here overlap with those referenced by Jan Kyncl.