>>96267129
You can only get speed once per turn, and only during your turn.
>As I play creatures, I lose 1 health and he does NOT get Start Your Engines counters.
Because it's not his turn
>When his creatures die I lose 1 hp and he gets an Engines counter. But as my creatures enter he still does not get a new counter.
He presumably attacked with a creature and it died, triggering bastion on his turn and thus getting speed. Curse still only triggers on your turn, so he doesn't get speed.

If you flashed in a creature on his turn, curse would trigger and he would get speed. If he blocked with a creature and it died, bastion would trigger on your turn and thus he wouldn't get speed.