Мастер сказал, что плинтус у двери нужно починить.

Not usually. In this kind of everyday sentence, мастер usually means something like repairman, technician, worker, or handyman, depending on context.

So Мастер сказал... is more like:

The repairman said...

rather than The master said...

Russian мастер can mean a skilled specialist in many contexts, so the exact English word depends on the situation.

Because что introduces a subordinate clause.

The sentence breaks into two parts:

• Мастер сказал = The repairman said
• что плинтус у двери нужно починить = that the baseboard by the door needs to be repaired

In Russian, subordinate clauses are normally separated by a comma, so the comma before что is required.

Because Russian past tense agrees with the subject in gender and number.

Here the subject is мастер, which is masculine singular, so the verb is сказал.

Compare:

• мастер сказал = the male worker said
• мастерица сказала = the craftswoman said
• мастера сказали = the workers said

Here у двери means by the door or near the door.

It describes the location of the baseboard:

• плинтус у двери = the baseboard by the door

Literally, у often means by, near, or at.

It does not mean that the door owns the baseboard. It is just a location phrase.

Because the preposition у requires the genitive case.

The dictionary form is дверь = door.
After у, it becomes двери:

• дверь → у двери

So:

• у двери = by the door

This is a very common pattern in Russian:

• у окна = by the window
• у стены = by the wall
• у дома = by the house

It actually is in the object form here, but for an inanimate masculine noun, the accusative singular is the same as the nominative singular.

So:

• nominative: плинтус
• accusative: плинтус

That is why it looks unchanged.

In the sentence, плинтус is the thing being repaired, so it is the direct object of починить.

A useful comparison:

• Я вижу стол = I see the table
  (стол stays the same because it is inanimate)

But:

• Я вижу кота = I see the cat
  (кот changes to кота because it is animate)

Because нужно + infinitive is a very common Russian way to express necessity.

• нужно починить = it is necessary to repair
• more natural English: needs to be repaired or has to be repaired

This structure is impersonal. It does not say who must do the action.

So the sentence means something like:

• The repairman said that the baseboard by the door needs to be repaired.

If you wanted to say who has to do it, Russian would usually make that explicit:

• Мне нужно починить плинтус = I need to repair the baseboard
• Ему нужно починить плинтус = He needs to repair the baseboard

This is about aspect.

• починить = perfective
  focuses on completing the repair successfully
• чинить = imperfective
  focuses on the process, repeated action, or general activity of repairing

In this sentence, the idea is that there is one specific repair that needs to be completed, so починить is the natural choice.

So:

• нужно починить = it needs to be fixed
• нужно чинить would sound more like it is necessary to be repairing it or one needs to repair it regularly / as an ongoing task, depending on context

No, it does not say.

That is one of the key things about нужно + infinitive here: the sentence gives the necessity, but not the person responsible.

So плинтус у двери нужно починить means:

• the baseboard by the door needs to be repaired

but it does not specify whether the repairman, the speaker, the listener, or someone else should do it.

If Russian wants to name the person, it often uses the dative:

• Мне нужно починить плинтус = I need to repair the baseboard
• Тебе нужно починить плинтус = You need to repair the baseboard

Yes. Russian word order is flexible, and changing it often changes emphasis rather than core meaning.

The original sentence is fairly neutral:

• Мастер сказал, что плинтус у двери нужно починить.

Other possible orders include:

• Мастер сказал, что у двери нужно починить плинтус.
  This puts a bit more focus on where the problem is.

• Мастер сказал, что нужно починить плинтус у двери.
  Also natural, with нужно починить presented earlier.

The basic meaning stays the same, but the information focus shifts slightly.

Yes. In most situations, надо and нужно are very close in meaning.

So this would also work:

• Мастер сказал, что плинтус у двери надо починить.

Very roughly:

• нужно can sound a little more neutral
• надо can sound a little more conversational

But in many everyday sentences, they are interchangeable.