>>2 depends on a lot of things.
But what do you mean by compete? I don't think you mean in Olympic games, gambling or drinking competitions, I think you mean compete in terms of population or in a 1v1 battle to the death, star-trek style.
now, if man is allowed to use man-made technology we beat all other lifeforms, hands-down.
T-Rex? machine gun
Pterodactyl? machine gun
Caveman? machine gun
Giant fish? machine gun
Jesus? machine gun
Godzilla? machine gun
If we aren't allowed to use anything other than our bare hands, then we may still win the majority of battles due to our intelligence.
I think that a modern day Tiger or Liger would beat the Sabretooth tiger; recent studies have shown that the Sabretooth tigers skull is not as capable of generating or withstanding as much force that a modern day tiger's can.
But remember, these creatures went extinct for a reason hence our modern day friends may have an advantage.
example:
horse vs raptor..3..2...1...GO!
The raptor stalks the horse, from downwind as to not alert it. As the horse lowers it's head to taste some delicious fresh grass the raptor closes in. The raptor picks up speed as the distance closes. The horse notices the raptor just in time. The horse whirls around and grabs the raptor between its mighty legs. It's a bearhug. The raptor can't breathe, it's ribcage shatters and the raptor drops to the floor, lifeless.
The horse rips out the raptors heart and feasts upon the bloody goodness.
VICTORY!111111111!!!