Paul téléphone pour prendre rendez-vous.

Here it is a verb, not the noun telephone/phone.

In Paul téléphone, téléphone is the third-person singular present tense form of téléphoner, meaning to phone or to call.

So the structure is:

• Paul = subject
• téléphone = verb

If it were the noun, you would normally expect something like le téléphone.

It is the present indicative.

More specifically, it is the il/elle/on form of téléphoner:

• je téléphone
• tu téléphones
• il/elle/on téléphone
• nous téléphonons
• vous téléphonez
• ils/elles téléphonent

Because Paul is one person, French uses the third-person singular form: téléphone.

Because pour + infinitive is a very common French pattern used to express purpose or goal.

So:

• Paul téléphone = Paul is calling
• pour prendre rendez-vous = in order to make an appointment

In other words, pour here means something like to or in order to.

French often uses pour + infinitive when the same person does both actions:

• Paul calls
• Paul makes the appointment

Since Paul is the one doing both things, the infinitive prendre is used.

Because prendre rendez-vous is the normal French expression. It is an idiomatic collocation.

Word for word, it looks like to take an appointment, but that is not how it should be translated into natural English. The natural meaning is:

• to make an appointment
• to arrange an appointment
• sometimes to book an appointment

This is a good example of why it is better to learn whole expressions, not just individual words.

Because prendre rendez-vous is a fixed expression, and it is normally used without an article.

So French usually says:

• prendre rendez-vous avec le médecin = to make an appointment with the doctor

not usually:

• prendre un rendez-vous

You may sometimes see an article in certain contexts, but the standard everyday expression is prendre rendez-vous.

Compare:

• Je prends rendez-vous demain. = I am making an appointment tomorrow.
• J’ai un rendez-vous demain. = I have an appointment tomorrow.

In the first sentence, it is the verbal expression prendre rendez-vous. In the second, un rendez-vous is a noun phrase meaning an appointment.

Because rendez-vous is a fixed expression that has become a standard noun/expression in French, and it is conventionally written with a hyphen.

You should simply learn it as:

• un rendez-vous
• prendre rendez-vous

It is also useful to know that the spelling stays the same in the plural:

• des rendez-vous

You can add the person or place after téléphoner, usually with à.

For example:

• Paul téléphone au médecin pour prendre rendez-vous.
• Paul téléphone à la clinique pour prendre rendez-vous.

That is because téléphoner normally takes à before the person or organization being called.

So:

• téléphoner à quelqu’un = to phone someone

Yes, often you could, but there is a small difference.

• téléphoner specifically means to phone
• appeler means to call

Examples:

• Paul téléphone au médecin.
• Paul appelle le médecin.

Notice the grammar difference:

• téléphoner à quelqu’un
• appeler quelqu’un

So with téléphoner, French uses à. With appeler, it does not.

Yes, very close.

• pour prendre rendez-vous = to make an appointment / in order to make an appointment
• afin de prendre rendez-vous = in order to make an appointment

Afin de is a bit more formal than pour. In everyday speech, pour is much more common.

You put ne ... pas around the conjugated verb:

• Paul ne téléphone pas pour prendre rendez-vous.

That means Paul is not calling to make an appointment.

Notice that ne ... pas goes around téléphone, not around prendre:

• Paul ne téléphone pas ...

because téléphone is the conjugated verb.

There are several common ways:

• Paul téléphone pour prendre rendez-vous ?
  Rising intonation in speech

• Est-ce que Paul téléphone pour prendre rendez-vous ?
  Very common spoken and written form

• Paul téléphone-t-il pour prendre rendez-vous ?
  More formal inversion

All three mean the same thing.