1) Generally, stiffer the coaxial cable, better it is. Best "cables" are actually solid copper tubes (usable to 100 GHz or so), so called rigid coax. Then there is semi-rigid coax (usable to 30 GHz or so) and finally, the ordinary "flexible" coax, usable maybe up to 1-2 GHz, depending on the length. Flexible coax exhibits changes in cable geometry when moving it around, so it might be unusable in reflection loss measurements if you need repeatability (see below). Since you mentioned BNC, that probably itself limits usable frequency range to less than 1 GHz, so you might get away with something like RG-179 (about same than average scope probe coax), or RG-58 (a bit thicker), if your cables are short. See some datasheets of the cables to get idea of the losses involved. Basically, thicker cables tend to have less loss.
2) 2-wire setup (i.e. loose wires) is usually not an option since impedance is completely unpredictable. For RF-design, things you want to measure are S-parameters, like input/output reflection loss (S11, S22) and transmission loss (S21, S12), For amplifiers, noise figure, intermodulation distortion (IP3, ACLR etc) and 1 dB compression point are common measurements. For this, you'll need tightly controlled impedance all the way. Probing is much more difficult with RF, since you really can't stick just about any kind of probe to your circuit without affecting internal matchings. Yes, there are microprobes and such things with controlled impedance all the way to the tip of the probe but these are not really things that average hobbyist will get.. So you must usually provide controlled impedance test points with proper coaxial connectors. Of course this depends your requirements, but that probably is more like worst case.
3) Measuring the impedance is somewhat difficult, and would involve a VNA. Basically, it is determined by the conductor geometry (thats why twisting the coaxial cable will change its impedance!) and permittivity of insulator dielectric.
Regards,
Janne