I, and many others, never understood the no pull rope thing....makes no sense
The relay and ski doo issue, does sound like a familiar issue though.
The other issue could be a gas line freeze.
Some heat, in a garage, would likely be best first choice to resolve.
When I have started my sleds in these temps, and colder temps if you can believe it, the first thing you had to do was get the engine pulling over somewhat easy, once you did that after a few easy & slow pulls on the rope, she would always fire right up.