Learn Russian: Дверь электрички чуть не прищемила мне руку. - The door of the commuter train almost caught my hand.

This is the standard Russian structure for an action that was on the verge of happening but was averted. The phrase чуть не acts as a fixed unit meaning 'almost'. Because the action ultimately did not happen, Russian includes the negative particle не. This structure is very common when a sudden, often negative, event is narrowly avoided.

Could I use the imperfective aspect here instead of the perfective прищемила?

No, to express an imminent averted action with чуть не or едва не, you must use a perfective verb in the past tense. The action of the door pinching the hand is viewed as a single, sudden result that was narrowly avoided, which makes the perfective aspect mandatory.

Why does the sentence use the dative pronoun мне instead of the possessive мою to say 'my hand'?

In Russian, when an action directly affects a body part, it is standard to put the person experiencing it in the dative case (мне) and the body part in the accusative (руку). Saying мою руку sounds unnatural in this context, almost as if the hand is a detached object you happen to own rather than a part of your body.

What kind of catching does прищемить describe? Could I use it for catching a ball?