Meine Mitbewohnerin steckt den Kassenbon an die Pinnwand, damit sie ihn nicht verliert.

Here is the case breakdown:

• Meine Mitbewohnerin = nominative
  It is the subject, the person doing the action.

• den Kassenbon = accusative
  It is the direct object, the thing being pinned up.

• an die Pinnwand = accusative after an
  An is a two-way preposition. Here it shows movement toward a destination, so it takes the accusative.

• sie = nominative
  Subject of the damit clause.

• ihn = accusative
  Direct object of verliert.

Also, meine has the ending -e because it is modifying a feminine singular nominative noun: Mitbewohnerin.

Because German uses two-way prepositions like an in two different ways:

• accusative for direction / movement toward
• dative for location / position

In this sentence, the receipt is being put onto the noticeboard, so there is a change of location:

• an die Pinnwand = onto the noticeboard

If you were just describing where it already is, you would use dative:

• Der Kassenbon steckt an der Pinnwand.
  = The receipt is attached to / on the noticeboard.

Here, stecken means something like:

• to put
• to stick
• to pin
• to attach

In this context, a natural English translation would be:

• She pins the receipt to the noticeboard
• She puts the receipt up on the noticeboard

So it is not just the literal English verb stick. It is a common German verb for placing something into, onto, or against something.

In this sentence, it is best understood as stecken plus the prepositional phrase an die Pinnwand.

So the structure is:

• steckt = verb
• den Kassenbon = object
• an die Pinnwand = where she puts it

That is different from the separable verb anstecken, which can mean things like:

• to infect
• to light
• to plug in
• sometimes to pin on / attach, depending on context

But here, the simplest reading is just stecken with an as a preposition.

Damit introduces a purpose clause.

Here it means:

• so that
• in order that

So:

• ..., damit sie ihn nicht verliert.
  = ..., so that she doesn’t lose it.

This clause explains the purpose of the action in the main clause.

Because damit sie ihn nicht verliert is a subordinate clause, and in German subordinate clauses are normally separated by a comma.

So the comma is required:

• Meine Mitbewohnerin steckt den Kassenbon an die Pinnwand, damit sie ihn nicht verliert.

Because damit introduces a subordinate clause, and in German subordinate clauses the finite verb goes to the end.

Compare:

• Sie verliert ihn nicht.
  Main clause: the verb is in second position.

• ..., damit sie ihn nicht verliert.
  Subordinate clause: the verb moves to the end.

That word order is very typical after conjunctions like:

• dass
• weil
• wenn
• damit

• sie refers to meine Mitbewohnerin
• ihn refers to den Kassenbon

Why?

• Mitbewohnerin is feminine singular, so sie matches it.
• Kassenbon is masculine, so the accusative pronoun is ihn.

Technically, sie could also match another feminine noun like die Pinnwand, but context makes that impossible here. A noticeboard is not the thing that might lose the receipt; the roommate is.

Because German pronouns follow the grammatical gender of the noun, not the natural gender and not whatever English would use.

• der Kassenbon = masculine
• accusative masculine pronoun = ihn

So:

• den Kassenbon → ihn

If the noun were neuter, then es would be possible. But Kassenbon is masculine, so ihn is correct.

Yes, absolutely.

You could say:

• Meine Mitbewohnerin steckt den Kassenbon an die Pinnwand, um ihn nicht zu verlieren.

That is also natural German.

Why does that work? Because the subject of both actions is the same person:

• she pins it up
• she does not want to lose it

A useful rule:

• Use um ... zu when the subject is the same.
• Use damit when the subject is different, or when you want to state the subject explicitly.

So in this sentence, damit and um ... zu are both possible.

Here, nicht is negating the whole idea of losing it.

In a subordinate clause, where the verb goes to the end, nicht often comes shortly before that final verb:

• ..., damit sie ihn nicht verliert.

Compare the main-clause version:

• Sie verliert ihn nicht.

So the position of nicht changes along with the normal word order of the clause.

Yes, some parts can move, but the basic grammar stays the same.

For example, you could front another element for emphasis:

• Den Kassenbon steckt meine Mitbewohnerin an die Pinnwand, damit sie ihn nicht verliert.

That is grammatically possible, but less neutral. The original sentence is the most natural everyday order:

• Meine Mitbewohnerin steckt den Kassenbon an die Pinnwand, damit sie ihn nicht verliert.

German allows some flexibility, but:

• the main clause still keeps the finite verb in second position
• the damit clause still sends the verb to the end