All three medians MUST lie inside the triangle.
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There can be no formula for the number of triangles within the polygon. Any triangle can be split into two by a line from one of the vertices to the opposite side. Thus the number of triangles can be increased by one by simply splitting one triangle. And then THAT triangle can be split in two, and the that one, ...
Yes. Any triangle can be inscribed within a circle, although the center of the circle may not necessarily lie within the triangle.
The answer will depend on how the triangle is situated within the rectangle (how many of the triangle's vertices coincide with those of the rectangle), and what other information you have.
An orthocenter on an obtuse triangle actually lies outside of the triangle. In an acute triangle, the orthocenter lies within the triangle.
There are three measurable angles within a triangle, and within a equilateral triangle there are thee equal angles, each measuring 60o .