Running speed, reload speed and pretty much everything is constant, and does not grow on its own.
It can be improved via Badass Ranks/Tokens by amount explicitly stated there.
Base health appears to follow the exponential function Health =
floor(80 * 1.13^level) In other words, each level increases your base
Health by 13% relative to the previous level. Equivalently, your base
Health doubles every 5.67 levels. At level 50, your base Health is
36058.
Source
Your damage does not increase itself, but there's a sliding scale that determines damage to targets of different levels. Thus, after you level up, you start to do more damage to a constant enemy of constant level. Here's the scale for BL1, I could not find one for BL2, maybe it's the same.