В отпуске я хочу больше гулять и меньше работать.

Because here в отпуске means on vacation / while on vacation, describing a state or location.

With в, Russian often uses:

• Accusative for movement into something:
  
  • в отпуск = on vacation in the sense of go on vacation
• Prepositional for being in that state/place:
  
  • в отпуске = on vacation, already there

Compare:

• Я еду в отпуск. = I’m going on vacation.
• Я в отпуске. = I’m on vacation.

So in your sentence, the speaker is already talking about what they want to do during the vacation, so в отпуске is correct.

Отпуске is in the prepositional case.

The dictionary form is отпуск. After в when it means in / during, Russian often uses the prepositional case:

• отпуск → в отпуске

This is a very common pattern:

• в доме = in the house
• в городе = in the city
• в отпуске = on vacation

So the -е ending shows that the noun is in the prepositional case.

This is just one of those places where Russian and English use different prepositions.

English says:

• on vacation

Russian says:

• в отпуске
  literally closer to in vacation

You should learn it as a fixed expression. Many everyday expressions do not match English word-for-word.

Not exactly. Гулять is broader than just to walk.

In this sentence, гулять can mean things like:

• to go for walks
• to stroll
• to spend time outside
• to go out and enjoy yourself

So больше гулять is often better understood as:

• walk more
• go out more
• spend more time outdoors

It is not the same as идти (to go, to be going) or ходить (to go habitually / on foot).
Гулять emphasizes leisure, relaxation, being out and about.

Because they depend on хочу (I want).

In Russian, after хотеть (to want), you normally use the infinitive:

• Я хочу спать. = I want to sleep.
• Я хочу читать. = I want to read.
• Я хочу гулять. = I want to walk / go out.

So:

• я хочу больше гулять и меньше работать
  = I want to walk/go out more and work less

Хочу is the 1st person singular form of хотеть (to want).

Conjugation:

• я хочу = I want
• ты хочешь = you want
• он/она хочет = he/she wants
• мы хотим = we want
• вы хотите = you want
• они хотят = they want

Since the subject is я (I), the verb must be хочу.

Yes. Here больше and меньше function as adverbs, modifying the actions:

• больше гулять = to walk/go out more
• меньше работать = to work less

They answer the question how much / to what extent?

These words are very commonly used with verbs:

• больше читать = read more
• меньше есть = eat less
• больше отдыхать = rest more

Because the comparison is general, not explicit.

Russian uses чем when you say more than or less than something specific:

• Я работаю больше, чем раньше. = I work more than before.

But in your sentence, there is no stated comparison like than before or than usual. It just means:

• more
• less

So больше гулять и меньше работать is completely natural without чем.

Because гулять and работать are two infinitives joined by и (and) as part of one structure after хочу:

• я хочу больше гулять и меньше работать

This is similar to English:

• I want to walk more and work less.

No comma is needed.

Yes, often you can.

Russian often drops subject pronouns when the verb form already makes the subject clear:

• В отпуске хочу больше гулять и меньше работать.

This still clearly means I want... because хочу already shows 1st person singular.

However, including я is also perfectly normal. It can make the sentence a bit clearer or more emphatic.

Yes, Russian word order is flexible.

The original order:

• В отпуске я хочу больше гулять и меньше работать.

puts в отпуске first, so it sets the scene: On vacation, ...

Other possible orders:

• Я хочу в отпуске больше гулять и меньше работать.
• Я в отпуске хочу больше гулять и меньше работать.

These are all understandable, but they may sound slightly different in emphasis.

A rough guide:

• В отпуске ... first = emphasizes during vacation
• Я хочу ... first = emphasizes what I want

Because the sentence talks about general ongoing activities, not a single completed action.

• гулять = to walk / stroll / spend time out
• работать = to work

The speaker is talking about how they want to spend their vacation in general:

• do more walking / leisure activity
• do less working

That naturally calls for the imperfective aspect, which is used for:

• repeated actions
• ongoing activities
• general habits
• process rather than completion

Not exactly.

Отпуск usually means vacation from work or leave for an adult employee.

Каникулы usually means school holidays or break.

So:

• Я в отпуске. = I’m on vacation / on leave.
• У детей каникулы. = The children are on holiday / break.

In your sentence, отпуск is the natural choice if an adult is talking about time off from work.

The stress is:

• в отпу́ске
• я хочу́
• бо́льше
• гуля́ть
• и
• ме́ньше
• рабо́тать

A simple pronunciation guide:

• В отпу́ске я хочу́ бо́льше гуля́ть и ме́ньше рабо́тать.

Approximate English-style help:

• v at-POOS-kye ya kha-CHOO BOL-she goo-LYAT' ee MEN-she ra-BO-taht'

The most important thing is to keep the stress in the right place, especially in:

• отпу́ске
• хочу́
• гуля́ть
• рабо́тать

Yes. Гулять is a very flexible verb.

Depending on context, it can mean:

• to walk
• to stroll
• to go out
• to spend time outside
• to celebrate / party
• to roam

For example:

• Мы гуляем в парке. = We’re walking in the park.
• Вчера они гуляли всю ночь. = Yesterday they partied all night.

In your sentence, because it is about vacation and contrasted with работать, the meaning is clearly something like walk/go out/spend time enjoying yourself.

A natural Russian version would be:

• Я хочу меньше работать и больше отдыхать.

Here отдыхать means to rest / relax.

That sentence is built exactly the same way as your original one:

• хочу
  • infinitives
• меньше / больше as adverbs

So your original sentence is a very useful pattern you can reuse.