Je fais la cuisine pour Paul.

Because faire la cuisine is a very common French expression meaning to cook or to do the cooking.

French also has the verb cuisiner, so you can say:

• Je fais la cuisine pour Paul.
• Je cuisine pour Paul.

Both are correct.
A small difference in feel:

• faire la cuisine often sounds like doing the activity of cooking / preparing a meal
• cuisiner is the more direct verb to cook

In everyday speech, both are natural.

Because faire la cuisine is a fixed expression. In this expression, French normally uses the article la.

So:

• faire la cuisine = to cook / to do the cooking

Without la, Je fais cuisine is not correct standard French.

This is one of those cases where French uses an article in a way that English does not.

Not here.

La cuisine can mean two different things depending on context:

• the kitchen
  
  • La cuisine est petite. = The kitchen is small.
• cooking / cuisine
  
  • J’aime la cuisine italienne. = I like Italian cuisine.
  • Je fais la cuisine. = I cook / I do the cooking.

In Je fais la cuisine pour Paul, it means cooking, not the room.

It can mean either one, depending on context.

The French present tense often covers both:

• I cook for Paul
• I am cooking for Paul

So Je fais la cuisine pour Paul could describe:

• a habit: I cook for Paul
• what is happening now: I’m cooking for Paul

If French wants to be extra clear that it is happening right now, it can say:

• Je suis en train de faire la cuisine pour Paul.
  = I am in the middle of cooking for Paul.

Pour Paul means for Paul.

It usually means the cooking is:

• intended for him
• done for his benefit
• prepared on his behalf

So the food is for Paul, or you are doing the cooking for him.

Because pour is the normal preposition for for in this meaning.

• pour Paul = for Paul

Using à Paul would usually suggest something more like to Paul or possession in another structure, not the idea intended here.

For example:

• Je donne le plat à Paul. = I give the dish to Paul.
• Je fais la cuisine pour Paul. = I cook for Paul.

So pour is the correct choice here.

Yes. That is completely natural.

• Je fais la cuisine pour Paul.
• Je cuisine pour Paul.

Both mean I cook for Paul.

A learner can think of it like this:

• faire la cuisine = a common expression
• cuisiner = the straightforward verb

If you want a simple version, Je cuisine pour Paul is great.

Fais is the present tense of faire with je.

The verb is faire = to do / to make

Present tense:

• je fais
• tu fais
• il / elle / on fait
• nous faisons
• vous faites
• ils / elles font

So Je fais la cuisine literally looks like I do the cooking, but in natural English it often becomes I cook.

Faire is a very common verb that can mean both to do and to make, depending on the expression.

Examples:

• faire ses devoirs = to do homework
• faire un gâteau = to make a cake
• faire la cuisine = to cook / to do the cooking

So here, you should learn faire la cuisine as a whole expression, not translate word by word too rigidly.

Pretty much, yes.

• Je = I
• fais la cuisine = cook / do the cooking
• pour Paul = for Paul

So the order is:

subject + verb expression + prepositional phrase

That is very similar to English: I cook for Paul.

A rough English-friendly pronunciation is:

zhuh feh la kwee-zeen poor pohl

A few notes:

• Je sounds like zhuh
• fais sounds like feh
• cuisine sounds like kwee-zeen
• pour sounds like poor
• Paul sounds like pohl

The s in fais is silent.

Yes, it sounds natural.

French speakers would understand it easily.
Also natural are:

• Je cuisine pour Paul.
• Je prépare à manger pour Paul. = I’m preparing food for Paul.

So your sentence is normal and idiomatic.

They are close, but not exactly identical.

• faire la cuisine = to cook / to do the cooking
• faire à manger = to make something to eat / to prepare food

Both can work in many situations, but faire la cuisine sounds more specifically like the activity of cooking, while faire à manger focuses more on preparing food for someone to eat.

For example:

• Je fais la cuisine pour Paul.
• Je fais à manger pour Paul.

Both are possible, but they are slightly different in nuance.