Wegen der Mieterhöhung müssen wir vielleicht umziehen.

Because wegen usually takes the genitive case in standard German.

• die Mieterhöhung = the rent increase, as a dictionary form
• after wegen → wegen der Mieterhöhung = because of the rent increase

Since Mieterhöhung is a feminine noun, its genitive singular article is der.

A useful pattern:

• wegen des ... for many masculine/neuter nouns
• wegen der ... for feminine nouns and plural nouns

In everyday spoken German, you may also hear wegen der/dem with dative in some regions, but wegen der Mieterhöhung is the standard form.

Wegen means because of, due to, or on account of.

So:

• Wegen der Mieterhöhung = Because of the rent increase / Due to the rent increase

It is a preposition, so it introduces a noun phrase rather than a full clause.

Compare:

• wegen der Mieterhöhung = because of the rent increase
• weil die Miete erhöht wurde = because the rent was increased

Both are possible, but wegen + noun is more compact.

Because German is a verb-second (V2) language in main clauses.

The first element in this sentence is:

• Wegen der Mieterhöhung

That whole phrase takes the first position. The conjugated verb must then come second:

• Wegen der Mieterhöhung müssen wir vielleicht umziehen.

So the structure is:

1. Wegen der Mieterhöhung
2. müssen
3. wir
4. the rest

If you start with the subject instead, you get:

• Wir müssen wegen der Mieterhöhung vielleicht umziehen.

That is also correct.

Because müssen is a modal verb, and modal verbs usually send the other verb to the end of the clause in the infinitive.

So:

• wir müssen umziehen = we have to move

Here:

• müssen is the conjugated verb
• umziehen is the infinitive that goes to the end

This is a very common pattern in German:

• Ich will gehen.
• Sie kann kommen.
• Wir müssen umziehen.

Because when a separable verb appears in the infinitive, it stays together:

• umziehen

It only splits when it is the conjugated main verb in a main clause:

• Wir ziehen um. = We are moving.

But with a modal verb:

• Wir müssen umziehen.

So:

• conjugated main verb: zieht ... um
• infinitive: umziehen

No. Umziehen can mean two different things:

1. to move house / relocate
2. to get changed / change clothes

Here it clearly means to move house, because the context is a rent increase:

• Wegen der Mieterhöhung müssen wir vielleicht umziehen.

If the context were clothes, you might see something like:

• Ich muss mich umziehen. = I need to get changed.

Context tells you which meaning is intended.

Vielleicht means maybe or perhaps.

In this sentence it modifies the whole idea:

• müssen wir vielleicht umziehen = we may have to move / we might have to move

Its position is natural and common, but adverbs like vielleicht can sometimes move around depending on emphasis.

For example:

• Wegen der Mieterhöhung müssen wir vielleicht umziehen.
• Wir müssen vielleicht wegen der Mieterhöhung umziehen.
• Vielleicht müssen wir wegen der Mieterhöhung umziehen.

All of these are possible, though they may sound slightly different in emphasis.

It is a compound noun, which is very common in German.

• die Miete = rent
• die Erhöhung = increase

Together:

• die Mieterhöhung = rent increase

In compounds, the last part determines the grammatical gender and basic meaning. Since Erhöhung is feminine, Mieterhöhung is also feminine:

• die Mieterhöhung

Because wir is the subject of the sentence: the people who may have to move.

• wir müssen ... umziehen = we have to move

German subject pronouns use the nominative case:

• ich, du, er, sie, es, wir, ihr, sie, Sie

Uns would be accusative or dative, not the subject form.

Yes, absolutely.

That version is also correct:

• Wir müssen wegen der Mieterhöhung vielleicht umziehen.

German often allows different elements to come first in a main clause. Whatever comes first is followed by the conjugated verb in second position.

So both are correct:

• Wegen der Mieterhöhung müssen wir vielleicht umziehen.
• Wir müssen wegen der Mieterhöhung vielleicht umziehen.

The first version puts more emphasis on the reason.

Yes, most likely. The phrase der Mieterhöhung uses a definite article, so it usually refers to a rent increase that is known or understood in the situation.

• wegen der Mieterhöhung = because of the rent increase

If you wanted a more indefinite idea, you might say:

• wegen einer Mieterhöhung = because of a rent increase

But in real life, rent increases are often specific, so der Mieterhöhung is very natural.

The basic meaning of müssen is must / have to.

But because the sentence also has vielleicht, the overall meaning becomes:

• we might have to move
• we may have to move

So:

• müssen = have to / must
• vielleicht müssen wir ... = we might have to ...

That is why English often uses might have to rather than a plain must here.