semi-regular polygons

[BMAD]

Define semiregular polygon as a polygon which has all of its' edges of the same length. Also, all of its' interior or exterior angles must be equal (meaning that any interior angle must be x or 360-x). It must be concave and simple (it should not self-intersect) and only two of its' edges are allowed to meet in each corner.

Find the semiregular polygon that has the minimum number of edges.

[k-man]

10-gon formed by 2 regular hexagons sharing a removed common edge.

[k-man]

12-gon shaped as an outline of a cross

[BMAD]

12-gon shaped as an outline of a cross

I have a smaller shape but this is a good start.

[bonanova]

It must be concave?

[BMAD]

yes.

[bonanova]

The vertices must lie on a grid.
That suggests a regular tiling of the plane: either triangle, square or hexagon.
Three triangles make a 9-gon, but six of those edges share a vertex. The OP does not allow that.
The 12-gon on a square tiling and 10-gon of the hex tiling seem to be all that are left.

Of those, the 10-gon wins.