Der Einzug ist für Anfang nächsten Monats geplant, der Auszug aber schon Ende dieses Monats.

They are nouns, so they are capitalized in German.

Both are event nouns built from verbs:

• einziehen → der Einzug
• ausziehen → der Auszug

German often uses nouns like this where English might also use a verbal phrase. In housing contexts, these nouns refer to the event of moving in or moving out.

Because both nouns are masculine:

• der Einzug
• der Auszug

There is no completely reliable shortcut here, so it is best to learn the noun together with its article. That said, many short noun forms created from verbs are masculine, but not all of them.

This is a case of ellipsis: German often leaves out repeated words when they are easy to understand from the first clause.

The fully expanded version would be:

Der Einzug ist für Anfang nächsten Monats geplant, der Auszug ist aber schon für Ende dieses Monats geplant.

In the original sentence, the repeated material in the second half is omitted because it would sound unnecessarily heavy. This is very normal, especially in written German.

Because it is also part of the ellipsis.

The second half is understood as:

der Auszug [ist] aber schon [für] Ende dieses Monats [geplant]

So für is logically there, even though it is not repeated. German often omits repeated function words too, not just repeated verbs.

aber marks a contrast: the first clause talks about the move-in, and the second contrasts it with the move-out.

schon adds the sense of already or as early as. It suggests that the move-out is happening sooner than one might expect, especially compared with the move-in.

So the feeling is roughly:

• move-in: beginning of next month
• move-out: already at the end of this month

Because with geplant, German very often uses für to mean scheduled for:

• für Montag geplant
• für nächste Woche geplant
• für Anfang nächsten Monats geplant

So für introduces the intended date or time.

You could also say am Anfang des nächsten Monats, but the version in the sentence is more compact and very natural in scheduling/planning language.

Yes, für does take the accusative, but it governs Anfang, not Monats.

The structure is:

• für
  • Anfang → accusative
• nächsten Monats → genitive phrase depending on Anfang

So the internal structure is:

für [Anfang [nächsten Monats]]

The same idea applies in the second half:

[für] Ende dieses Monats

Here Ende is the noun after für, and dieses Monats is genitive depending on Ende.

Because Monat is a masculine noun, and in the genitive singular it normally adds -s:

• der Monat
• des Monats

Since both Anfang and Ende are followed by a genitive expression, you get:

• nächsten Monats
• dieses Monats

They are different kinds of modifiers, but both are shaped by the same grammatical environment: genitive masculine singular because of Monats.

• dieses Monats: dieses is the determiner form of dieser
• nächsten Monats: nächsten is the adjective form of nächste

If you included an article in the first phrase, you would get:

• Anfang des nächsten Monats

So:

• nächsten Monats = no article, adjective directly before the noun
• dieses Monats = demonstrative determiner before the noun

Because German main clauses normally place the finite verb in second position, while other verbal material often goes later.

Here the pattern is:

• ist = finite verb, in second position
• geplant = participle, at the end

So:

Der Einzug | ist | für Anfang nächsten Monats | geplant.

This is a very common structure in German: etwas ist für X geplant = something is planned/scheduled for X.

For a learner, the most useful way to understand it is simply as a common expression meaning is planned or is scheduled.

Grammatically, it is close to a passive-like or adjectival participle construction. In everyday use, though, you usually do not need to worry much about the label. Just learn the pattern:

• Der Termin ist für Freitag geplant.
• Der Einzug ist für Anfang nächsten Monats geplant.

Yes.

Common alternatives include:

• Anfang nächsten Monats
• Anfang des nächsten Monats
• am Anfang des nächsten Monats

and:

• Ende dieses Monats
• am Ende dieses Monats

The version in your sentence is concise and natural, especially in written or administrative language. The alternatives are not wrong; they are just a little fuller or differently styled.

Because they focus on different parts of the moving process.

• der Umzug = the move as a whole
• der Einzug = the move-in
• der Auszug = the move-out

So this sentence is specifically contrasting the two separate events: moving out happens first, moving in happens later. Using Umzug would not make that contrast as clearly.