Yes. Medians always intersect in a single point, called the centroid, or geocenter.
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A triangle has by definition three intersecting sides. If two of the sides are parallel, they will never intersect, so no triangle can ever be formed.
Perpendicular lines intersect at a 90 degree angle. Parallel lines do not intersect, nor would they ever intersect if continued infinitely.
Sure. If one of the base angles is more than 90 degrees, then the altitude (height) is outside the triangle. Yes. This only occurs with an obtuse triangle. Because an altitude is a line drawn from a vertex to the opposite side and is perpendicular with that opposite side, it can only occur if it is outside the triangle. Look at the triangle in related links. If you look at the vertex on the top, the only way to draw the altitude would be to draw outside the triangle.
It is obtuse
No it can't. The hypotenuse of a right triangle will always be longer than either one of the other two sides.