Horizontal or vertical polarization of an cubical QUAD-Antenna is determined by placement of the feed point (feed gap on the driven element)
Parasitic elements need to be mounted in the same orientation as the driven element. If tuning gap is present on the parasitic elements, the placement of this gap does not effect the polarization.
Feed line is made out of standard coaxial cable (RG8/RG58) 50ohm unbalanced, to attach this feed line to the balanced drive element (Quad loop) without proper matching is not advisable, instead use an 1:1 balun, (however I rarely use a balun ;)
16 gauge to 10 gauge wire will work best. Insulated wire will work fine. There is no need to adjust the calculations for using insulated wire. The velocity ratio is close to 99% and because of the adjustments needed because of the antenna surroundings, the measurements calculated should be really be close enough to start your adjustments. Using THHN wire may have a higher resonant frequency, adding 1% to the lenghts when using THHN wire should work well.
The gain of an antenna with parasitic elements varies with the spacing and tuning of the elements, and thus for any given spacing there is a tuning condition that will give maximum gain this spacing. The maximum front to back ratio seldom, if ever occurs at the same condition that gives maximum forward gain. The impedance of the driven element also varies with the tuning and spacing, and thus the antenna system must be tuned to its final condition before the match between the coax and the antenna can be completed. However, the tuning and matching may interlock to some extent, and it is usually necessary to run through the adjustments several times to insure that the best possible tuning has been obtained.