Ich nehme eine Rolle Toilettenpapier aus dem Schrank.

Nehme is the 1st person singular present-tense form of nehmen:

• ich nehme = I take / I am taking
• du nimmst = you take
• er/sie/es nimmt = he/she/it takes

So because the sentence starts with Ich, the verb has to be nehme.

• nehmen is the infinitive: to take
• nimm is the imperative: take!

Because Rolle is a feminine noun in German: die Rolle.

That means in the accusative singular it becomes eine Rolle.

Here it is the direct object of nehmen:

• Ich nehme was? → eine Rolle Toilettenpapier

So eine is used because:

• the noun is feminine
• it is singular
• it is in the accusative

In eine Rolle Toilettenpapier, the word Toilettenpapier works like a substance or mass noun after a quantity/container word.

This is very common in German:

• ein Glas Wasser = a glass of water
• eine Tasse Kaffee = a cup of coffee
• eine Scheibe Brot = a slice of bread
• eine Rolle Toilettenpapier = a roll of toilet paper

So German usually does not add an article there. English uses of, but German often just puts the noun directly after the measure word.

Yes. German often expresses of relationships without a separate word like of.

So:

• eine Rolle Toilettenpapier literally looks like a roll toilet paper
• but it means a roll of toilet paper

This pattern is normal and natural in German.

Because the preposition aus always takes the dative case.

The noun Schrank is masculine:

• nominative: der Schrank
• dative: dem Schrank

So:

• aus + dem Schrank = out of the cupboard / closet / cabinet

That is why you get aus dem Schrank, not aus den Schrank.

Because with aus, the case is determined by the preposition itself, not by the idea of movement.

Some German prepositions change case depending on location vs direction, but aus is always dative.

So:

• aus dem Schrank = out of the cupboard

Even though there is movement, aus still requires dative.

They mean different kinds of movement:

• aus dem Schrank = out of the cupboard / from inside the cupboard
• vom Schrank = off the cupboard / from the cupboard

So if the toilet paper was inside the cupboard, aus dem Schrank is correct.

If something were sitting on top of the cupboard and you took it from there, vom Schrank might make sense.

German word order is flexible, but this is a very normal order:

• Ich = subject
• nehme = verb in second position
• eine Rolle Toilettenpapier = object
• aus dem Schrank = prepositional phrase showing where from

Putting aus dem Schrank at the end sounds natural and neutral.

You could also say:

• Ich nehme aus dem Schrank eine Rolle Toilettenpapier.

That is also grammatical, but it shifts the emphasis a bit more toward where you are taking it from.

It can mean either one, depending on context.

German present tense often covers both:

• I take
• I am taking

So Ich nehme eine Rolle Toilettenpapier aus dem Schrank could mean:

• I take a roll of toilet paper out of the cupboard
• I am taking a roll of toilet paper out of the cupboard

Context tells you which is meant.

Yes, and that is also very natural.

There is a slight difference:

• nehmen = to take
• holen = to get / fetch

So:

• Ich nehme eine Rolle Toilettenpapier aus dem Schrank focuses on taking it.
• Ich hole eine Rolle Toilettenpapier aus dem Schrank often sounds more like getting it from there.

In many everyday situations, both are possible.

The plural of Rolle is Rollen.

So:

• eine Rolle Toilettenpapier = one roll of toilet paper
• zwei Rollen Toilettenpapier = two rolls of toilet paper

Notice that Toilettenpapier stays the same here, because it is a mass noun.

German often does not need a separate word for of in expressions of quantity, containers, and portions.

Compare:

• a glass of water → ein Glas Wasser
• a cup of tea → eine Tasse Tee
• a piece of cake → ein Stück Kuchen
• a roll of toilet paper → eine Rolle Toilettenpapier

So this structure is something worth getting used to, because it is very common.

Schrank is a general word for a cupboard, cabinet, or wardrobe, depending on context.

So aus dem Schrank could mean:

• out of the cupboard
• out of the cabinet
• out of the closet

The best English translation depends on what kind of Schrank it is in the real situation.