Der Einzug ist erst im Juni geplant, der Auszug aber schon Ende Mai.

In this context:

• der Einzug = the move-in / moving in
• der Auszug = the move-out / moving out

They come from the verbs:

• einziehen = to move in
• ausziehen = to move out

So the sentence is talking about the planned move-in date and move-out date.

Because they are nouns, and all nouns are capitalized in German.

These words are related to verbs, but here they are being used as things/events:

• einziehen → der Einzug
• ausziehen → der Auszug

English does not do this in the same way, so this often stands out to learners.

Because both Einzug and Auszug are masculine nouns:

• der Einzug
• der Auszug

There is no special reason in this sentence beyond normal noun gender. As in all German, the article has to match the noun’s grammatical gender.

No. Here erst does not mean first.

In this sentence, erst means something like:

• not until
• only as late as

So:

• Der Einzug ist erst im Juni geplant
  = The move-in is not planned until June

This use of erst is very common when something happens later than expected or later than another thing.

Here schon means:

• already
• as early as

So:

• der Auszug aber schon Ende Mai
  = but the move-out already at the end of May

The contrast erst ... schon ... is very common in German:

• erst im Juni = not until June
• schon Ende Mai = already at the end of May

It emphasizes that one thing is later, while the other is surprisingly early.

Because im is a contraction of in dem:

• in dem Juni → im Juni

With months, German often uses in + dative to mean in a certain month:

• im Mai
• im Juni
• im Oktober

So im Juni simply means in June.

German often uses fixed time expressions without a preposition where English needs one.

So:

• Ende Mai = at the end of May
• Anfang Juni = at the beginning of June
• Mitte Juli = in the middle of July

This is a very common pattern. You should usually learn Ende Mai as a set time expression.

Because the second part is really a shortened version of a full clause.

The full version would be:

• Der Einzug ist erst im Juni geplant, der Auszug ist aber schon Ende Mai geplant.

In that full clause:

• der Auszug = first element
• ist = second element
• aber comes later

Then German leaves out the repeated ist geplant, because it is understood:

• der Auszug aber schon Ende Mai

So aber is not strange here at all. It belongs to the second clause, which has been shortened by leaving out repeated words.

Because German often avoids repeating words that are already clear from context.

The sentence could be written in full as:

• Der Einzug ist erst im Juni geplant, der Auszug ist aber schon Ende Mai geplant.

But that sounds more repetitive. German commonly omits the repeated part when the meaning is obvious.

This is called ellipsis: leaving out words that do not need to be repeated.

The comma separates two main clauses:

• Der Einzug ist erst im Juni geplant
• der Auszug aber schon Ende Mai

Even though the second clause is shortened, it is still understood as a separate clause, so the comma is correct.

The first clause follows normal German main-clause word order:

• Der Einzug = subject
• ist = finite verb in second position
• erst im Juni = time expression
• geplant = predicate adjective/past participle at the end

So:

• Der Einzug | ist | erst im Juni | geplant

This is very typical German structure: the finite verb is in position 2, and the rest of the predicate often comes later.

Yes. A fuller version would be:

• Der Einzug ist erst im Juni geplant, aber der Auszug schon Ende Mai.
• Der Einzug ist erst im Juni geplant, der Auszug ist aber schon Ende Mai geplant.

These all express basically the same idea.

The original version is natural and compact. It sounds smooth because German often removes repeated material when it is obvious.