Yes.
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An infinite number in a Euclidean plane - which is the "normmal" plane. Some selected numbers in the finite or affine planes (but you need to be studying projective geometry to come across these).
They came from geometry. If you have a square whose sides are 1 unit long then its diagonal is sqrt(2) units long.
Yes.
geo- & -metry or -metr meaning earth an measure respectively. Latin
Platonic solids, being three-dimensional objects with certain characteristics, would come under the category of geometry, not algebra. For more information on Platonic solids, see the Wikipedia entry or the Wolfram MathWorld description.