Installing sub-panel, plus hot tub to new sub
Number 6 AWG can be protected at 70 amps if the equipment at both ends is rated for 75°C yes to your wire colors.
Yes
If the new sub is attached to the main or right next to it, it does not need a ground rod. The NEC requires a grounding electrode at a separate building or structure containing more than one branch circuit. The single circuit to the hot tub does not require a ground rod. The equipment goundung conductor with the circuit is sufficient.


Nipples between panels of 24" or less do not require you to derate the wire ampacity.

When attaching a sub-panel I would not limit myself to 1" conduit. I would use a 1 1\2" or 2" nipple to allow more wire to pass between the panels. Just a personal preference though.

Good luck and stay safe!