What's embarrassing is that you haven't even bothered to research what these words actually described in the KJV times.
Unicorn = a single-horned rhinoceros
Cockatrice = a poisonous snake
Satyrs = symbolic of pagan religions (a demonic wasteland - a place void of God's goodness)
Dragons = dinosaurs or dinosaur-like creatures
Your take on cockatrice is correct, but I'll have to disagree with you in regards to the animals referred to by Unicorn, Satyr, and Dragon.
First, Unicorn. It is almost indisputable that the original referent was clearly an animal with two horns, if only because the same hebrew word is also used in Deuteronomy 33, and it is explicitly said there that the animal has multiple horns. The one horn idea seems to come originally and exclusively from the Septuagint, which uses the greek word monokeros. It's not clear why this term is used, again, because the monokeros of Deut 33 clearly has more than one horn. Either the Greek-speaking Hebrews meant something else by monokeros (perhaps metaphorical, rather than a literal description of the referrant), or they simply got it wrong. In any case, the KJV can't be wholly blamed for this, because they were just essentially transliterating the Latin (unicornis), derived from the Greek (monokeros).
As for satyr, the Hebrew word used is שָׂעִיר, which roughly transliterates into English as sa'ir. I doubt this is coincidence. It's hard to know the etymology of the greek word satyrus, but given sa'ir is actually a word for goat in Hebrew (I think it literally means something along the lines of 'hairy thing'), it's not unreasonable to think that the Greek actually derives from the Hebrew somewhere along the line. Regardless, the referrant is not as metaphorical as being symbolic of pagan religions - it means goats, either literally as in that Isaiah passage, or elsewhere as a specific pagan practice of goat-idol worship (I think sa'ir also appears earlier in the Bible, maybe Exodus, and there is translated in the KJV as goat). In either case, for whatever reason, it seems most likely that the KJV writers transliterated the Hebrew here, rather than actually translating it.
As for dragons, no need to posit dinosaurs. It could be a sea monster of some sort, but the root world is almost unanimously translated as serpent in Exodus, including by the KJV. Big serpent-y monster, maybe. Either way, the idea that it could mean a dinosaur is one that is a possibility, but largely one that has been imported into the text, not derived from it.
