In the shack, does it MATTER what 50 ohm coax you use for interconnecting within the system?
Generally, no. If the cables are short, it's insignificant. Even if you use RG-58, the loss in a 10 foot section at 29 MHz is 0.26 dB. At 3.9 MHz, the loss is only 0.08 dB. RG-8X is a better choice for higher power. Here the loss is only 0.16 dB at 29 MHz and only 0.05 dB 3.9 MHz.
To be more precise, is it best to use only one cable from the same reel, say like RG-213 or whatever your favorite flavor? Does velocity factor even matter on these short runs?
Unless you are dealing with a phased array antenna system, velocity factor does not matter at all.
Last, is there a preferred cable type by a certain manufacturer that is considered to be the best for the kind of operating we do?
No. If the cable is required to make sharp bends, you should probably stay away from some foam core types because of center conductor migration. Otherwise, use stuff that can handle the power you plan to run and mate with good connectors properly installed.