There's no method to climbing up ledges, it's normal physics, if throw yourself forward you move forward, you can use the analogue to move in air slightly. Half your body is a spring joint that acts as your legs, you can just slide over the ledge but you have to put the ledge low enough.
Just do what you would in real life to climb up something, you put your arms below you and crouch down to lift your legs up.
I don't think a lot of people understand what I'm domnstrating here, there are no game mechanics, I didn't program climbing, or pole vaulting, or rope swinging, these are just side effects of physically attached hands, all I did was make your hands attach things correctly to your character body, the rest just happens as a result.
About the running, there is no top speed, that's why jogging more vigorously results in faster movement. Walking is a problem, see my newest video for a glimpse of the hybrid system.
Jumping is something that confuses everybody, you don't have to jump off the ground at all, I'm literally just tiptoeing quickly, so when you say you're landing before the in game jump, well that's because you're lifting off the ground.
Using the Trigger to grab things was necessary, as it's preparing you for my actual game SLAYR, the grips control picking up and reloading weapons, while triggers just grab things naturally, can't use the fire trigger to pickup and drop a weapon.