I looked up some typical parts for 10.7 MHz IF use. For example, the Toko
154AC-470052N0 ( **broken link removed** ) is a metal can with an adjustable "core", probably made of ferrite or powdered iron. It appears that the core is actually not just a core but a small core and a large bobbin, both ferrite, with the bobbin designed to adjust up or down on the outside of the coil. There is at least one capacitor inside, in parallel with primary or secondary. Might be one on each, I can't tell.
Wire diameter is probably quite small magnet wire, I'm guessing based on past experience, but likely in the range of 28 to 34 AWG. Number of turns is tailored to provide the specified transformation ratio (you can buy many different ratios) while also resonating the inductance with the parallel cap.