Wenn mein Taschenrechner nicht funktioniert, rechne ich die Aufgabe lieber langsam auf Papier nach.

Because wenn introduces a subordinate clause in German. In subordinate clauses, the conjugated verb usually goes to the end.

• Main clause: Mein Taschenrechner funktioniert nicht.
• With wenn: wenn mein Taschenrechner nicht funktioniert

So this is a normal word order pattern:

• wenn + subject + other elements + verb

This is one of the first big word-order differences English speakers notice in German.

Because the sentence begins with the subordinate clause, and that whole clause takes up the first position in the sentence. In a German main clause, the conjugated verb must come second.

So the structure is:

• Wenn mein Taschenrechner nicht funktioniert, = position 1
• rechne = position 2
• ich = subject after the verb

That is why you get:

• Wenn mein Taschenrechner nicht funktioniert, rechne ich ...

and not:

• Wenn mein Taschenrechner nicht funktioniert, ich rechne ...

It can mean either if or when, depending on context. In this sentence, English could naturally translate it as:

• If my calculator doesn’t work, ...
• or When my calculator doesn’t work, ...

German wenn is often used for:

• conditions = if
• repeated situations in time = when/whenever

So here it has a general conditional meaning: whenever that situation happens, the speaker does this.

Nachrechnen is a separable verb. It means something like:

• to check by calculating
• to calculate again
• to work out again in order to verify

In a main clause, separable prefixes split off and go to the end:

• infinitive: nachrechnen
• main clause: ich rechne ... nach

That is exactly what happens here:

• rechne ich die Aufgabe lieber langsam auf Papier nach

So nach belongs with rechne, even though it appears at the end.

Die Aufgabe is in the accusative case because it is the direct object of nachrechnen.

The speaker is recalculating the problem/exercise/task.

• ich rechne die Aufgabe nach

Here:

• ich = subject
• rechne ... nach = verb
• die Aufgabe = direct object

Since Aufgabe is feminine, die is the same in both nominative and accusative singular, so the form does not change.

In this context, Aufgabe most likely means:

• problem
• exercise
• math problem
• assignment/task

Because the sentence talks about a calculator and recalculating something, math problem or exercise is the most natural interpretation.

So die Aufgabe nachrechnen means something like to work through the problem again.

Yes, lieber here means rather or preferably.

It expresses preference:

• Ich rechne die Aufgabe lieber langsam auf Papier nach.
• I’d rather work the problem out slowly on paper.

A useful thing to know is that lieber is related to gern:

• gern = gladly / with pleasure
• lieber = rather / prefer
• am liebsten = most preferably / best of all

So lieber adds the idea that, in this situation, doing it slowly on paper is the preferred option.

Here langsam functions as an adverb, modifying the verb phrase.

It tells you how the speaker recalculates the problem:

• langsam = slowly

German often uses the same form for adjectives and adverbs:

• adjective: ein langsamer Rechner = a slow calculator
• adverb: ich rechne langsam = I calculate slowly

So there is no special -ly ending as in English.

Auf Papier is an idiomatic way to mean on paper in the sense of using paper as the medium.

It is similar to English:

• I wrote it on paper
• meaning not digitally, but physically on paper

If you say auf dem Papier, that usually sounds more like:

• on the paper
• referring to a specific piece of paper

So in this sentence:

• auf Papier = on paper, by hand, using paper

That is the more natural choice.

Yes, Taschenrechner is a compound noun:

• Tasche = pocket/bag
• Rechner = calculator/computer, literally something that calculates

So Taschenrechner literally suggests a pocket calculator.

Its gender is masculine:

• der Taschenrechner

That is why the sentence says:

• mein Taschenrechner

not meine or mein Taschenrechnerin, etc.

Yes. You could say:

• Wenn mein Taschenrechner nicht funktioniert, dann rechne ich die Aufgabe lieber langsam auf Papier nach.

The dann is optional here. It can make the conditional relationship a little clearer or more emphatic, similar to English then.

Without dann, the sentence is still completely natural:

• Wenn ..., rechne ich ...

With dann, it sounds slightly more explicit:

• Wenn ..., dann rechne ich ...

Yes. German often uses the present tense for general habits, repeated situations, and real conditions, just like English can.

This sentence describes a typical reaction:

• if the calculator does not work, the speaker recalculates the problem on paper

So the present tense is exactly what you would expect:

• funktioniert
• rechne

It does not have to mean only right now. It can also mean whenever this situation happens.

Yes, German word order is somewhat flexible, but some versions sound more natural than others. The original sentence is very natural:

• rechne ich die Aufgabe lieber langsam auf Papier nach

You could also move some adverbial elements around, for example:

• rechne ich die Aufgabe auf Papier lieber langsam nach
• rechne ich lieber die Aufgabe langsam auf Papier nach

But these versions may change emphasis and can sound less smooth.

A helpful default pattern for learners is:

• verb + subject + object + adverbs/details + separable prefix

So the original sentence is a very good model to follow.