Presumably I lost the pet would be Saj vIchIltaH,...
You don't really want the
-taH continuous idea. Using it implies an ongoing process of misplacing the pet, whereas you probably want to indicate a state of its having been misplaced.
Saj vIchIl is fine for
I lost the pet, and you can say
Saj vIchIlpu' if you really need to show that the act of losing it is complete.
But then what about I am lost, I could try I lose myself - jIchIl'eghtaH
We have two official ways to say
I am lost:
jIHtaHbogh naDev vISovbe' I don't know the here I am being (this one is difficult to translate)
DaqwIj vISovbe' I don't know my locationAnd similarly Get lost! could be the imperative yIchIl'egh.
I usually just say
naDevvo' yIghoS go away from here.