Parameters of a metal detector coil
A larger diameter coil will produce the same magnetic flux density as a smaller coil but at a greater distance. This is important. So a 20cm coil is better than a 10cm coil.





The symmetry is such that all the terms in this element are constant except the distance element dL, which when integrated just gives the circumference of the circle. The magnetic field is then: -



As distance from the plane of the current loop increases, z dominates and flux density tends to be an inverse cube law with distance Z. Data/pictures taken from here

As for the wire, there will be a pulse of current associated with the applied voltage pulse and this may be in the order of an amp so choose an appropriate wire gauge so that dc losses are not significant.

Regards insulation, basic copper wire with enamel is perfectly fine.