A Bigfoot-like ancestor...
Well firt off, look at my name!

I believe that if Ice Age hunters didn't hunt them, Woolly Mammoths and mabye Elasmatheriums would still be alive. I think it would be cool if they survived but I doubt it.
The dinosaurs are definetly Extinct. Their relatives may still be living in the form of birds and reptiles but they are, unfortunetly, a long gone great race of reptilians.
I don't know much about the stories of Steller Sea Cows or Honshu Wolves (or the other wolf subspecies native in to Japan that's still said to be alive) so I can't really say weither or not I think they're still alive or not. It would be amazingly awesome if they managed to survive (especially the Steller Sea Cow).
Caribbean Monk Seals probably are Extinct but there's good video evidence (or is it something else...).
I believe when people find Big Cat tracks on Javan, they are finding Javan Leopard tracks and mistaking them for Javan Tigers unfortunetly. I would love nothing more them for all tiger subspecies to still be out and about. I believe a few Caspian Tigers may have survived until today out in the far deserts of Iran and such.
Living in a town that has seen cougars every now and then and having an incident with a large black cat down my very stree, I believe that some Eastern Cougars still exsist. Most Cougar sightings are probably escaped animals form private collections or the occasional wanderer (although I don't think any wanderer has made it down to Connecticut) but there still are those few. It is believed that a relic population of 50 lives in the Canadian portion of the Appalachian Mountains.
Um it is actually widely believed that Benjamin was a female because of the lack of male genetalia in all photos and video evidence and without a body it can never be proven. I don't know if this part of the story is true, but I heard that after "Ben" died they threw his/her body in the dumpster and destroyed the exhibit.