Врач сказал, что мне нужно лечиться дома и больше отдыхать.

Because нужно often appears in an impersonal construction in Russian.

• мне нужно лечиться = I need to get treatment / I need to recover
• Literally, it is closer to to me, it is necessary to get treatment

So мне is in the dative case, because it marks the person for whom something is necessary.

Compare:

• Я должен лечиться = I must get treatment / I am obliged to get treatment
• Мне нужно лечиться = I need to get treatment
  This sounds a bit less about duty and more about necessity.

So in this sentence, мне нужно is the natural way to say I need to.

Here нужно is not a normal adjective. It is a predicative word meaning necessary / needed.

That is why it stays the same:

• мне нужно
• ему нужно
• нам нужно

It does not agree like an adjective would.

Compare:

• нужный врач = a needed/necessary doctor → adjective
• мне нужно отдыхать = I need to rest → predicative word

So in this sentence, нужно simply means it is necessary.

Because лечить and лечиться mean different things.

• лечить = to treat/cure someone
• лечиться = to be treated / to undergo treatment / to get medical treatment

Examples:

• Врач лечит пациента. = The doctor treats the patient.
• Пациент лечится дома. = The patient is being treated at home / is recovering at home.

In your sentence, the speaker is the person receiving treatment, not the person giving treatment, so лечиться is correct.

Not usually in a literal English sense.

Even though -ся often gets called reflexive, in many verbs it does not mean the person is literally doing the action to themselves. With лечиться, it usually means:

• to receive treatment
• to be under treatment
• to recover / get medical care

So мне нужно лечиться дома is better understood as:

• I need to be treated at home
• I need to stay home and recover
• I need to undergo treatment at home

It does not necessarily mean the person is personally acting as their own doctor.

Because after нужно, Russian normally uses an infinitive to say what action is necessary.

Structure:

• кому + нужно + infinitive

Examples:

• Мне нужно работать. = I need to work.
• Ему нужно спать. = He needs to sleep.
• Нам нужно ехать. = We need to go.

So here:

• мне нужно лечиться дома
• и больше отдыхать

Both verbs stay in the infinitive because they describe the actions that are necessary.

Because дома is the normal adverb meaning at home.

• дома = at home
• домой = homeward / to home
• в доме = in the house/building

So:

• Я дома. = I am at home.
• Я иду домой. = I am going home.
• Я в доме. = I am inside the house/building.

In your sentence, the idea is not specifically inside a building, but rather at home rather than elsewhere, so дома is the natural word.

It means to rest more.

Here больше is functioning like more, modifying the verb отдыхать.

• отдыхать = to rest
• больше отдыхать = to rest more

This can mean:

• more often
• for longer
• to a greater extent

It does not mean the bigger rest or anything like that. It is simply the usual Russian way to say rest more.

Because Russian can use one нужно for both infinitives when they are part of the same idea.

So:

• мне нужно лечиться дома и больше отдыхать

means:

• I need to get treatment at home and rest more

You could repeat it for emphasis:

• мне нужно лечиться дома и нужно больше отдыхать

But that sounds less smooth unless you want special emphasis. Normally, the single нужно covers both actions.

Because что introduces a subordinate clause after сказал.

Main clause:

• Врач сказал = The doctor said

Subordinate clause:

• что мне нужно лечиться дома и больше отдыхать = that I need to get treatment at home and rest more

In Russian, a comma is normally required before что in this kind of sentence.

So:

• Врач сказал, что...

is the standard punctuation.

Yes, Russian word order is fairly flexible, though some versions sound more natural than others.

Original:

• Врач сказал, что мне нужно лечиться дома и больше отдыхать.

Possible variations:

• Врач сказал, что дома мне нужно лечиться и больше отдыхать.
• Врач сказал, что мне дома нужно лечиться и больше отдыхать.

These are understandable, but the original is very natural and neutral.

The original order works well because:

• мне нужно introduces the necessity clearly
• лечиться дома stays together naturally
• и больше отдыхать adds the second recommendation smoothly

So yes, the order can change, but the given sentence is a good standard version.

Because the sentence is talking about a general process or ongoing recommendation, not a single completed action.

• лечиться imperfective = to be undergoing treatment / to recover
• отдыхать imperfective = to be resting / to rest regularly

The doctor is not saying:

• do one single finished act of treatment
• rest once and be done

Instead, the doctor is giving advice about an ongoing situation:

• stay under treatment
• rest more over a period of time

That is why imperfective is natural here.

Yes, but it changes the tone.

Compare:

• Мне нужно лечиться дома и больше отдыхать.
  = I need to get treatment at home and rest more.

• Я должен лечиться дома и больше отдыхать.
  = I must / am supposed to get treatment at home and rest more.

нужно sounds more like necessity or medical need.

должен sounds more like:

• obligation
• duty
• requirement

In a doctor’s advice, нужно is often the more natural and less harsh choice.

Врач specifically means a medical doctor/physician.

It does not mean any health worker in general. For example:

• врач = doctor
• медсестра = nurse
• медик = medical worker/medic

So Врач сказал... clearly means The doctor said...

A useful breakdown is:

• Врач сказал = The doctor said
• что = that
• мне нужно = I need to / it is necessary for me
• лечиться дома = to be treated / recover at home
• и больше отдыхать = and rest more

So the sentence has this pattern:

• [Someone said] + что + [person in dative] + нужно + [infinitives]

This is a very common Russian structure, and it is worth learning as a chunk:

• Мне нужно...
• Тебе нужно...
• Ему нужно...
• Нам нужно...