Frankly, I'm surprised these animations are so bad. Sorry. Particularly, the jumping one - it looks like he's just suddenly being sucked upward by an invisible force, and certainly not moving by himself. Get up. Now jump up as high as you can.
-You totally forgot recoil. When you jumped just now, you had to "charge" your legs in a spring action so you could launch yourself upward. I'd say this is the biggest mistake. Toss in some recoil and you'll be on the right track.
-This character is supposed to be funny, right? Exaggerate his movements for that cartoony funny effect. What, don't you watch any cartoons? You may need to go do some "field research". Make his legs kick as if swimming while in the air, like Luigi in Mario Bros 2. Think of other lovable characters, like Earthworm Jim - Such a rememberable character because he had so much expressive character in his movements. Oh yeah, and he had some great idle animations, too. The best sprite chars have great idle animations. Your caveman guy could grab his knees and pant for breath, being overweight as he is. You could even code some animation swaps so his idle anim used is contingent on how much he's been running.
-When descending, give some indication of falling with the hair flying up.
-His mouth is distracting, not sure why he's opening it like that.
Hope my drivel is helpful.