Yesterday, I had a game where I had Dragon Tempest on the board, and I played a flying dragon, saying that it gets haste. At the moment when I wanted to deal 1 damage thanks to the trigger, my opponent used Green Slime to negate it. I understand that it was his mistake because, in the end, he destroyed my Dragon Tempest, but my dragon still gets haste?
That's a communication issue. Your opponent should have made it clear, which trigger he's countering. And you should have made it clear, that there are two triggers. But yes, if your opponent counters the damage trigger, your dragon still gets haste. If he counters the haste trigger, your dragon still gets to deal damage. Either way, Tempest is destroyed.
Note, that while both triggers are on the stack, your opponeng can counter either one. He does not have to wait for the top one to resolve before responding to the bottom one. So if the damage trigger is on top, for example, and you try to resolve it, your opponent flashing in the Slime does not mean, that he has to counter that one. He can counter the bottom trigger even with the top one still on the stack.
Just to make sure. So, no matter which trigger gets countered, I will be able to use the one that wasn't countered, and only after that, the enchantment gets destroyed?
The Enchantment is destroyed as part of countering the ability. But yes, you get the uncountered ability. Once an ability exist on the stack it doesn't matter what happens to the source. The ability will resolve regardless.