Paul est sûr.

Because the subject is Paul, which is he in grammatical terms: third person singular.

The verb être in the present tense goes like this:

• je suis
• tu es
• il / elle / on est
• nous sommes
• vous êtes
• ils / elles sont

So with Paul, you need est:

• Paul est sûr.

You would use es with tu:

• Tu es sûr.

Because sûr is an adjective here, not a noun.

In Paul est sûr, the structure is:

• Paul = subject
• est = verb to be
• sûr = adjective describing Paul

This works just like English Paul is sure, where sure is also an adjective and does not need an article.

The circumflex accent (^) is part of the spelling of sûr.

In modern French, it often helps distinguish sûr from sur:

• sûr = sure, certain
• sur = on, on top of

So the accent is very useful because it separates two different words.

In pronunciation, the circumflex does not always create a dramatic difference for learners, but you should learn the correct spelling because it matters in writing.

A simple learner-friendly pronunciation is:

• Paul ≈ pol
• est ≈ eh
• sûr ≈ a rounded vowel sound that English does not really have, followed by an r

Roughly:

• Pol eh sur
  but with a French u sound in sûr, not an English oo

A more IPA-style pronunciation is:

• Paul /pɔl/
• est /ɛ/
• sûr /syʁ/

Also, the t in est is normally silent here.

Yes. French adjectives usually agree with the person or thing they describe.

Forms of sûr:

• masculine singular: sûr
• feminine singular: sûre
• masculine plural: sûrs
• feminine plural: sûres

Examples:

• Paul est sûr.
• Marie est sûre.
• Paul et Jean sont sûrs.
• Marie et Anne sont sûres.

The extra letters are often not strongly pronounced, but they must be written correctly.

Yes, depending on context.

It can mean:

• sure / certain
• safe / reliable / secure

In Paul est sûr, the most natural reading is usually Paul is sure / certain.

If you want to say safe, French often uses related expressions depending on context, such as:

• en sécurité = safe
• sans danger = not dangerous
• un endroit sûr = a safe place

So sûr can cover more than one idea, but context tells you which one is meant.

Because after être, French normally uses an adjective directly to describe the subject.

This is a very common pattern:

• Paul est sûr.
• Marie est contente.
• Le livre est intéressant.

So this is not about adjective order like a big house or a red car. It is a predicate adjective: it comes after être, just as in English Paul is sure.

Normally, no.

In Paul est sûr, the t in est is silent because sûr begins with a consonant sound.

So you do not pronounce it like estt.

However, in some cases before a vowel sound, French can make a liaison:

• il est arrivé → the t can link to the next word

But in Paul est sûr, no liaison happens.

Yes, often you can.

• Paul est sûr.
• Paul est certain.

Both can mean that Paul is certain.

That said, sûr is extremely common in everyday French.
Also, French often uses sûr with de or in expressions like:

• Paul est sûr de lui. = Paul is self-confident
• Je suis sûr que... = I am sure that...

So sûr is a very useful word to learn early.

Yes. Very often, sûr is used with de.

Examples:

• Paul est sûr de sa réponse. = Paul is sure of his answer.
• Je suis sûr de cela. = I am sure of that.
• Elle est sûre d’elle. = She is sure of herself / self-confident.

So Paul est sûr is a complete sentence, but sûr can also introduce more information with de.