Hooking up hot tub

Just pulled a permit to install the used hot tub i just got. Straight shot the tub is only about 20 feet from the breaker box we have in the garage (200 amp service), but it will be around 30 feet by way of EMT conduit. The hot tub takes a 60 amp, 4-wire service and if the run is anything over 20 feet use 6 guage wire instead of 8 guage. The plan is to run 6 gauge THNN from the 220 breaker in the house panel, through EMT conduit inside using just the set screw couplings, through the wall to the outside where i will put a spa GFCI breaker (at least 5 feet away and within sight of the hot tub.) outside I will use EMT with push lock couplings. Now my first question. I dont think the electrical connectors on the hot tub are big enough for 6 gauge. I know your not supposed to go DOWN in wire size in the middle of a run, but since the 60 AMP GFCI will be outside and fed with 6 gauge, can I run 8 gauge out of the GFCI and into the hot tub? Like I said It would only be around a 10 ft run. Second question. Im having a hard time finding both 6/4 and 8/4 THHN that has the 2-hott, 1-neutral and 1-ground in it. They have 6/4 and 8/4 THHN single wire of all the different cables/colors though. Can I just get the single wires and run them all through the same conduit?

Thanks guys, I just want to have all the bases covered before I start and before i have it inspected when im done. Im located in Michigan.