How does the shape of the coil relate to the magnetic field around it?
Yes, the shape of the coil affects the shape of the field.

The most general answer to your question is that each tiny increment of wire produces a circular field around itself, and the overall field around the coil (or any length of wire of any shape) is the sum (integral) of all of those incremental fields. This is not a simple problem to solve in the general case, but we can make some general statements.

In a coil, the current in adjacent turns is going to have the same magnitude, and produce essentially the same field. This means that the space directly between the turns will have zero net field, since the contributions from the wires on either side of that space have opposite signs and cancel each other out.

This means that the strongest field will be found along the surface formed by the overall colleciton of turns. For a helical coil, this would be on the inside and outside surfaces of the cylinder formed by the turns, and for a planar (spiral) coil, this would be on either side of the plane.