Yes. Hledat = to look for, to search for. "Hledáte" is the 2nd person, plural.
The definition given in the hover text is merely "looking." It would be more helpful if it said "looking (for)"
yes, 'hledět' means 'look at something/someone', 'hledat' means 'look for'
What in the Czech sentence requires that the preposition be "for" rather than "at"?
At least as it's been used in the course so far, the verb "hledat" itself means to "look FOR." (I don't know yet if it also means "to look" in general...) For "to look at," the verb "dívat se (na)" is used. So, to answer your question more directly, I'd say that it's the verb in the Czech sentence that determines the preposition used in the English translation.
Disclaimer: I am learning, too...
Toho is shown as the or those Koně is shown as horse or horses. How can we tell which to use?
Toho is always singular (masculine or neuter, here accusative case). So it cannot be plural "horses".
hledate has no translation while pushing the finger/mouse button over it.