Learn Swedish: Läraren tittade på eleven med pekfingret mot dörren. - The teacher looked at the student with his index finger towards the door.

This is a descriptive absolute construction. It uses med (with) followed by a noun phrase (pekfingret) and a directional expression (mot dörren). In Swedish, this is a very common way to describe someone's body language or posture while they are performing the main action (like looking).

Why does the sentence use the definite form pekfingret instead of a possessive pronoun like hans or sitt?

In Swedish, when describing body parts that obviously belong to the person doing the action, you normally use the definite article instead of a possessive pronoun. Because it is part of this med construction, med pekfingret naturally translates to "with his index finger" without needing a possessive word.

Could I use till dörren instead of mot dörren here?

No, mot is the much better choice here. Mot means "towards" or "in the direction of," which fits a pointing gesture perfectly. Till (to) implies an actual destination that something is traveling to and arriving at, which does not work well for a stationary finger pointing from across the room.

Grammatically, is it possible that the student is the one pointing at the door?