После работы я хочу сделать паузу и отдохнуть.

Because после always requires the genitive case.

• The dictionary form is работа (nominative).
• The genitive singular of работа is работы.

So:

• после работы = after work
• после урока = after the lesson
• после фильма = after the movie

You cannot say после работа; that would be ungrammatical.

Работы is genitive singular.

For many feminine nouns ending in -а in the nominative:

• Nominative: работа
• Genitive singular: работы

Other examples:

• книга → книги (нет книги – there is no book)
• машина → машины (ключ от машины – the key of the car)

A simple rule to remember here:
после + [genitive], so when you see после, expect the next noun in genitive.

The difference is aspect:

• сделать паузу – perfective (focus on the result, one completed action)
• делать паузу – imperfective (focus on process / repeated action)

In this sentence, you mean I want to take a (one) break, a single completed act, so perfective сделать is natural.

Compare:

• Я хочу сделать паузу. – I want to (take) a break (once, now).
• Я люблю делать паузу каждый час. – I like taking a break every hour (habitually, repeatedly).

Again, this is aspect:

• отдохнуть – perfective, “to get (some) rest, to have rested” (completed result)
• отдыхать – imperfective, “to rest / to be resting” (process)

With хочу, both are possible, but they mean slightly different things:

• Я хочу отдохнуть. – I want to rest (and finish resting, feel rested).
• Я хочу отдыхать. – I want to be in a state of resting / to spend time resting (e.g., on vacation, in general).

In your sentence, сделать паузу и отдохнуть fits well because both verbs point to completed actions during that break.

The verb хотеть is typically followed by an infinitive:

• я хочу сделать – I want to do
• я хочу отдохнуть – I want to rest

Pattern:
[subject] + хотеть + infinitive

So:

• Я хочу есть. – I want to eat.
• Я хочу спать. – I want to sleep.
• Я хочу поехать. – I want to go (by transport).

In your sentence, there are two infinitives linked by и:

• сделать (паузу) and отдохнуть
  both depend on хочу (I want to take a break and to rest).

Паузу is accusative singular of the feminine noun пауза.

For many feminine nouns in -а:

• Nominative: пауза
• Accusative singular (direct object): паузу

You use the accusative case for the direct object of the verb:

• сделать что? – паузу
• читать книгу
• купить машину

So сделать паузу = “to make (do) a pause / break” where паузу is the thing being “made” (direct object).

Russian word order is relatively flexible. These variants are all possible and natural:

• После работы я хочу сделать паузу и отдохнуть. (your version – neutral)
• Я хочу после работы сделать паузу и отдохнуть.
• Я хочу сделать паузу и отдохнуть после работы.

The meaning is basically the same. Minor differences:

• Putting после работы at the beginning sounds like you’re setting the time frame first (After work, I want…).
• Putting после работы at the end can slightly emphasize when you want to rest.

Yes. In Russian, subject pronouns are often omitted when the verb ending makes the subject clear.

• (Я) хочу сделать паузу.
• (Я) устал.

So:

• После работы хочу сделать паузу и отдохнуть. – completely natural in speech and writing.

Including я is also fine; it may sound a bit more explicit or slightly more emphatic on I.

Here и is just the normal coordinating conjunction and linking two infinitives that share the same subject and verb хочу:

• я хочу [сделать паузу] и [отдохнуть]

It works exactly like English “and” in:

• I want to take a break and rest.

Using a comma here would be incorrect; Russian uses a conjunction (like English) to join parallel infinitives in such a structure.

Yes, сделать перерыв is also possible, but there is a nuance:

• пауза – a pause, often shorter, can be in speech, in music, or just a small break in activity.
• перерыв – a break, often regular/scheduled or more “official”: lunch break, lesson break, etc.

Examples:

• Сделать паузу и помолчать. – To pause and be silent.
• Сделать перерыв в работе. – To take a break from work (often longer, more formal).

In your sentence, both сделать паузу and сделать перерыв work; пауза can sound a bit more casual or “small rest,” but context decides.

Stresses (the stressed vowel is in capitals in transliteration):

• после – по́сле – PO-sle
• работы – рабо́ты – ra-BO-ty
• я – я – ya
• хочу – хочу́ – ha-CHU
• сделать – сде́лать – SDYE-latʹ
• паузу – па́узу – PA-u-zu
• и – и – ee
• отдохнуть – отдохну́ть – at-dakh-NUTʹ

Correct stress is important because it often changes the vowel quality in unstressed positions.

Yes, and it is perfectly natural.

• После работы я хочу… – After work I want…
• Когда я закончу работу, я хочу… – When I finish work, I want…

Both express the time relationship.
После работы is shorter and more typical; когда я закончу работу makes the idea of finishing more explicit but doesn’t change the basic meaning in this context.