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Q: Why do certain shapes tessellate or tile and others do not?

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A regular hexagon will tessellate.

Yes

Yes because it will tessellate

Some shapes tile because they fit together.

A regular octagon will not tessellate but an irregular one can.

Yes a triangle will tessellate leaving no gaps or overlaps

Sort of. Triangle is one of the shape that may tessellate(fit tightly). But one may argue that triangle is not a square (for sure), nor a tile (yet I think it may be) to a tessellate.

The rectangle is the simplest and most obvious case of a geometrical form that can tile a plane.

I would think that the obvious answer would be yes. maybe I am missing something in the question why would one think that you could not tile a seven sided polygon? * * * * * Yes, maybe you are missing something - mathematic reality. There are no polygons - regualr or irregular of 7 or more sides which will tessellate with identical shapes.

some shapes tile and some don't because if the side angles don't equal a factor of 360 degrees it will not tile and if the side angles do equal a factor a 360 degrees it willtile.

heptagon and an equilateral triangle

If it's interior angle is a factor of 360 then it will tessellate such as a square, a regular hexagon and an equilateral triangle.

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