Learn Russian: Кто-то наступил мне на ногу и поставил огромный синяк. - Someone stepped on my foot and gave me a huge bruise.

This is the Dative of inalienable possession. In Russian, when an action happens to a body part, you usually do not use a possessive pronoun like мою. Instead, you put the affected person in the Dative case (мне) and just name the body part. It literally means 'Someone stepped to me on the foot'.

Why is it на ногу (Accusative) instead of на ноге (Prepositional)?

The verb наступить (to step) implies motion directed onto a surface. Because the foot is the destination of the stepping action, Russian requires на + Accusative case, rather than the Prepositional case which only describes static location.

Why is the verb поставил (placed/stood) used for giving a bruise?

This is a standard Russian collocation. While English uses the verb 'to give', Russian treats a bruise as a physical mark that is 'placed' or 'set' on the skin. The phrase поставить синяк is the normal, everyday way to express this.

Does the мне apply to the second part of the sentence about the bruise too?

Yes. The Dative мне establishes that you are the 'victim' of this whole sequence of events (the Dative of detriment). Because the verbs наступил and поставил share the same subject and form a single connected event, мне carries over to the second verb automatically without needing to be repeated.