See https://en.wikipedia.org/wiki/Voiceless_palatal_stop It (Ť) is an unvoiced Ď, not D.
You can search for recordings of Czech words at forvo.com.
Yes. BUT... there is a way (at least for a Czech native speaker) to tell the difference. If it were the foxes looking for the dogs, not only would it be PSY, but TI would become TY
Lišky hledají ty psy.
If we omitted the TI-TY, nobody could be sure who was looking for whom and the general understanding would be that the first noun is an object and thus the foxes are the ones looking.
There is no way to figure this as a listening only exercise: Foxes and dogs sound the same in both Nominative and Accusative cases. While foxes can only be written one way, dogs can be either psi or psy, which sound the same. So this could be "Lišky hledají ty psy" Meaning that the fox is the one acting (by exclusion, because the dog is in accusative case)