I found the hardest part with the cones was actually spotting them coming! In PGR2 they were highly visibly cones, now they are just sticks!
Anyway, here is my advise for what it may be worth:
Learn the track and try to remember where the cones are. Most of the time it is left, right, left, right etc, but I found it would be where two cones would be on the right and break the left, right pattern where I would be thrown.
Take your time, there is not that much of a time rush even with Platinum. It is more important to go relatively slowly, ensuring you get every gate than to be flying down the road.
Don't use a mental ultra fast car. I was trying to get them using my beloved RUF RTurbo but it would occasionally spin out or I would end up going just too fast. I then changed to the Ultima GTR, which was fantastic for the challenges. It has great acceleration to boost you to the next cone in time if you do end up going quite slow, it is also very controllable and stable and a little tweak of the handbrake will bring the back end out nicely, with enough power to create a slide on demand. And is does not end up going insanely fast!!
As for 360's I never did any, just tried to pull of a reasonable slide here and there on the big turns and applying a bit of handbrake here and there to bring the back end round on the tighter cones.