Learn Russian: Я оступился, чуть не упал, и теперь у меня болит нога. - I stumbled, almost fell, and now my leg hurts.

In Russian, the phrase чуть не combined with a perfective past tense verb expresses an action that was on the verge of happening but didn't. You can think of it literally as meaning that by a little bit, you did not fall. The не is mandatory here to show the action was successfully averted.

Could I use the imperfective verb падал here instead of the perfective упал?

No, the чуть не construction requires a perfective verb when talking about a sudden, averted action in the past. If you used the imperfective падал, it would awkwardly imply that you were in the ongoing process of almost falling over a period of time, which doesn't fit a quick stumble.

Why does оступился have the reflexive -ся suffix? Does it mean I tripped myself?

Not exactly. While -ся often means doing something to oneself, here it simply makes the verb intransitive, meaning it cannot take a direct object. The root verb ступить means to step, and the prefix о- often indicates a mistake or missing the mark. Together, оступиться means to make a false step.

Just as a refresher, why is нога in the nominative case in у меня болит нога?