Mijn nek en arm zijn moe van het vasthouden van de ladder, maar de plank hangt eindelijk goed.

Both are possible.

Dutch often uses the possessive just once when it clearly belongs to both nouns:

• mijn nek en arm
• mijn nek en mijn arm

The version with one mijn is a bit more compact and natural in many contexts. Repeating mijn can add a little emphasis or clarity, but it is not necessary here.

Because the subject is actually two things together: nek + arm.

Even though nek and arm are each singular, mijn nek en arm is a coordinated subject, so Dutch treats it as plural:

• Mijn nek en arm zijn moe.

Compare:

• Mijn nek is moe.
• Mijn arm is moe.
• Mijn nek en arm zijn moe.

Because moe is a predicate adjective here: it comes after the verb zijn.

In Dutch, predicate adjectives normally do not take the -e ending:

• De arm is moe.
• De nek en arm zijn moe.

You only usually get the -e ending when the adjective comes before a noun:

• de moeë arm is not normal Dutch
• more natural: de vermoeide arm

So in this sentence, moe stays just moe.

Literally, it means something like from the holding of the ladder.

In natural English, you would usually understand it as:

• from holding the ladder
• because of holding the ladder

So the whole part moe van het vasthouden van de ladder means:

• tired from holding the ladder

The two vans do different jobs.

• The first van means from / because of and explains the cause of being tired.
• The second van belongs to het vasthouden van de ladder, literally the holding of the ladder.

So structurally it is:

• moe van ... = tired from ...
• het vasthouden van de ladder = the holding of the ladder

In English, the second of is often not translated directly, because we prefer holding the ladder rather than the holding of the ladder.

Because vasthouden is being used as a noun here.

Dutch can turn an infinitive into a noun-like expression by putting het in front of it:

• het eten = eating / the eating
• het werken = working
• het vasthouden = holding

This is very common in Dutch. So het vasthouden van de ladder means holding the ladder.

Vasthouden is a separable verb.

In the infinitive, it is written as one word:

• vasthouden

But when it is conjugated in a main clause, it usually splits:

• Ik houd de ladder vast.
• Hij houdt de plank vast.

So:

• infinitive: vasthouden
• finite main clause form: houdt/houd ... vast

That is why you see het vasthouden here as one word.

Yes. In Dutch, hangen can mean not only to hang, but also to be hanging / to be suspended / to be positioned hanging.

So de plank hangt goed means something like:

• the shelf/board is hanging properly
• the shelf is finally positioned correctly

This is very natural in Dutch when talking about something attached to a wall or placed so that it hangs straight.

Because goed is not describing plank directly before the noun. Instead, it describes how it hangs.

So here goed works like an adverb or predicative complement:

• De plank hangt goed. = The shelf hangs well / is hanging correctly.

But before a noun, you would use goede:

• de goede plank = the good plank

Compare:

• een goede plank = a good plank
• de plank hangt goed = the plank is hanging properly

That is a normal Dutch word order for adverbs in the middle/end part of the sentence.

Here:

• eindelijk means finally
• goed gives the result or manner: properly / correctly

So de plank hangt eindelijk goed means:

• the shelf is finally hanging properly

The position sounds natural because eindelijk modifies the whole situation, and goed stays close to the verb phrase it belongs with.

You may also hear other natural variations depending on emphasis, such as:

• De plank hangt nu eindelijk goed.
• Eindelijk hangt de plank goed.

But the original word order is perfectly normal.

Yes, you could, and many learners find door easier to understand.

• moe van het vasthouden van de ladder = tired from holding the ladder
• moe door het vasthouden van de ladder = tired because of holding the ladder

Both are understandable and natural.
Van often sounds a bit more idiomatic with moe:

• moe van iets
• moe van het wachten
• moe van het tillen

So the sentence as given is very normal Dutch.