Даже занятый человек должен отдыхать.

Даже means even.

In Даже занятый человек должен отдыхать, it emphasizes занятый человек:
Even a busy person must rest.

Russian word order is flexible, but placement changes what feels emphasized. For example:

• Даже занятый человек должен отдыхать. = Even a busy person must rest.
• Занятый человек даже должен отдыхать. = This sounds less natural here and shifts the emphasis.
• Человек должен отдыхать даже занятый... = not correct for this meaning.

So in this sentence, putting даже before занятый человек is the natural choice.

Because it agrees with человек.

Человек is:

• masculine
• singular
• nominative case

So the adjective must also be:

• masculine
• singular
• nominative

That gives занятый.

Compare:

• занятый человек = a busy man / person
• занятая женщина = a busy woman
• занятое место = an occupied seat
• занятые люди = busy people

Because человек is the subject of the sentence.

The sentence is about who must rest:
занятый человек = the busy person

In Russian, the subject is usually in the nominative case. So:

• занятый человек = nominative subject
• должен отдыхать = must rest

If the phrase had a different role in the sentence, the case would change. For example:

• У занятого человека нет времени. = A busy person has no time.
  Here занятого человека is in the genitive.

Должен agrees with the subject человек, which is masculine singular.

Forms of должен:

• должен = masculine singular
• должна = feminine singular
• должно = neuter singular
• должны = plural

So:

• Человек должен отдыхать. = A person must rest.
• Женщина должна отдыхать. = A woman must rest.
• Люди должны отдыхать. = People must rest.

Why not должен быть?
Because должен here already means must / should. It is followed directly by an infinitive:

• должен отдыхать = must rest

Должен быть would mean must be, which is a different structure:

• Он должен быть дома. = He must be at home.

It is the short form of должный, but in modern Russian it is mostly used as a modal word meaning must / should / is supposed to.

For learners, the easiest way to understand it is:

• должен + infinitive = must / should + verb

Examples:

• Я должен работать. = I must work.
• Ты должен учиться. = You should study.
• Человек должен отдыхать. = A person must rest.

So although it comes historically from an adjective, in real usage it behaves like a modal expression.

Because отдыхать is the imperfective verb, and it fits the meaning here.

• отдыхать = to rest, to be resting, to rest regularly/in general
• отдохнуть = to have a rest, to rest and finish resting once

In this sentence, the idea is general:
a busy person needs rest as a regular thing. That is why Russian uses the imperfective:

• должен отдыхать = must rest / should get rest

If you said:

• должен отдохнуть

that would usually mean must get some rest now / must rest once and recover.

So:

• Человек должен отдыхать. = general truth, regular need
• Ты должен отдохнуть. = you need to rest now / take a break

Yes, Russian word order is flexible, but different orders sound different in emphasis.

The most neutral and natural version here is:

• Даже занятый человек должен отдыхать.

Other possible orders:

• Даже человек занятый должен отдыхать.
  Possible, but less neutral and more marked.
• Занятый человек тоже должен отдыхать.
  This means A busy person also must rest, which is slightly different from even.
• Отдыхать должен даже занятый человек.
  This strongly emphasizes отдыхать or the whole statement.

So yes, word order can change, but the original sentence is the best basic version for a learner.

Russian has no articles.

So человек can mean:

• a person
• the person
• sometimes just person/people in general, depending on context

In this sentence, English usually translates it as:

• Even a busy person must rest.

That sounds natural because the sentence expresses a general truth. Russian does not need an article to show that.

It can mean more than one thing, depending on context.

Common meanings of занятый:

1. busy

   • Я сейчас занят. = I’m busy now.

2. occupied / taken

   • Это место занято. = This seat is occupied.
   • Телефон занят. = The phone line is busy.

In занятый человек, the meaning is clearly busy person, not occupied person.

You would change занятый and должен so they agree with the subject.

Examples:

• Даже занятая женщина должна отдыхать.
  Even a busy woman must rest.

• Даже занятое существо должно отдыхать.
  Even a busy being/creature must rest.
  (grammatically correct, though less common)

• Даже занятые люди должны отдыхать.
  Even busy people must rest.

Agreement is very important in Russian, especially with adjectives and words like должен.

Yes, that is also a very natural sentence:

• Даже занятому человеку нужно отдыхать.

It means almost the same thing:
Even a busy person needs to rest.

But the structure is different:

• Даже занятый человек должен отдыхать.
  занятый человек is the subject, in nominative.

• Даже занятому человеку нужно отдыхать.
  занятому человеку is in the dative case, because Russian often uses
  кому? нужно + infinitive = someone needs to...

This second structure can sound a little softer or more natural in everyday speech, while должен can sound stronger, more like must.

The stressed syllables are:

• Да́же
• заня́тый
• челове́к
• до́лжен
• отдыха́ть

So you can read it approximately as:

Да́же заня́тый человек до́лжен отдыха́ть.

A rough English-style pronunciation guide would be:

DA-zhe zan-YA-tïy che-la-VYEK DOL-zhen at-dï-KHAT'

That is only approximate, but the stress placement is important, especially in words like занятый and отдыхать.