Start with the cone generated by revolving a equilateral triangle about 
a median.

Take three such cones.

Arrange them so their points form the vertices of a equilateral triangle 
and their bases intersect in a line normal to this equilateral triangle.

Does the resulting solid have 3 vertices or 5?

If the triangle used to generate each cone has edge length 1, what is 
the edge length of the triangle formed by the points of the cones? I 
suspect it might be 1.5 as their bases have dihedral angles of 60 
degrees, so their diameters in the plane of the large triangle form the 
long diagonals of a hexagon, implying an overlap equal to the radius of 
the cone bases.

An infinite family can be formed by n cones formed by revolving 
isosceles triangles with an unequal angle equal to the internal angle of 
a regular n-gon... however, my suspicion is that 4 90-degree cones would 
have bases that reach a neighbor's apex, and that 5 or more would leave 
central gaps...

Take the bicone formed by revolving a square about a diagonal. What do 
you get by intersecting two whose axes are at 90-degrees? 3 bicones? 4 
or more with non-orthogonal axes?

What does the family of solids formed by revolving regular n-gons and 
regular star polygons about all of their lines of symmetry and taking 
the intersection? Is their any pattern to the shapes?