Translation:Are they searching for the cat? It is over there.
You can just say "there" for that in English too. "Over" is only added for emphasis and extra clarification. Tuossa could be "right there" too, but in English, we're not forced to distinguish between "right there" and "over there".
tässä = (right) here
täällä = (over) here
tuossa = (right) there
tuolla = (over) there
Duolingo accepted "Etsivätkö he kissa se on tuolla".
Is the non partitive allowed here? or is this duolingo letting 'small mistakes' slip through? I try to follow the Duolingo advice to say the sentence aloud after - but if I'm repeating/learning incorrect grammar, there's not much point in continuing with Duolingo - this seems to be happening quite a lot.
No. The direct object of "search" is the place where the searching takes place. What the subject is trying to find is introduced with for.
They are searching the forest [object = location] for the missing child.
The police searched my flat [object = location] for the murder weapon
I searched my cat [object = location] for fleas.
I searched the park [object = location] for my cat
You can drop either one. "I searched the park" - you don't know what I was looking for. "I searched for my cat" - you don't know where I searched.
So "Are they searching the cat" means "Are they looking for something on the cat?" (or as KristianKumpula suggested: in the cat)