Paul accepte avec plaisir.

Accepte is the third-person singular present form of accepter.

So:

• j'accepte = I accept
• tu acceptes = you accept
• il/elle/on accepte = he/she/one accepts

Since Paul is third-person singular, French uses accepte.

Because French spelling often shows verb endings that are silent in pronunciation.

In accepte, the final -e is not pronounced. It is there to mark the conjugation in writing.

For example:

• j'accepte
• tu acceptes
• il accepte

These forms sound very similar, but they are spelled differently.

Not necessarily in a short sentence like this.

In real French, accepter often does take an object:

• Paul accepte l'offre. = Paul accepts the offer.

But the object can be left unstated if it is already understood from context:

• someone makes a proposal
• then you say Paul accepte avec plaisir

So this can mean something like:

• Paul accepts it
• Paul agrees gladly

French often leaves this kind of thing implicit when the context is clear.

Avec plaisir is an adverbial expression meaning something like:

• gladly
• willingly
• with pleasure

It tells you how Paul accepts.

So:

• Paul accepte. = Paul accepts.
• Paul accepte avec plaisir. = Paul accepts gladly / with pleasure.

Because avec plaisir is a fixed idiomatic expression.

In this expression, French normally uses no article:

• avec plaisir = with pleasure / gladly

You would not normally say avec le plaisir here.

French has many expressions like this where English learners may expect an article, but French does not use one.

Because that is the normal, neutral word order in French.

French often puts short adverbial expressions after the verb:

• Paul accepte volontiers.
• Paul accepte avec plaisir.

You can move avec plaisir for emphasis in some contexts, but the basic sentence is most naturally:

• Paul accepte avec plaisir.

A careful pronunciation is approximately:

/pɔl aksɛpt avɛk plɛziʁ/

A rough English-friendly guide:

• Paul ≈ pohl
• accepte ≈ ak-SEPT
• avec ≈ ah-VEK
• plaisir ≈ pleh-ZEER with a French r

A few important points:

• the l in Paul is pronounced
• the final -e in accepte is silent
• the final r in plaisir is pronounced in French

No special liaison is added here.

You simply pronounce:

• Paul
  • accepte

Since the l in Paul is already pronounced, it just flows naturally into the next word:

• Paul accepte → pɔl aksɛpt

So this is not like cases where a normally silent final consonant appears because of liaison.

Yes. A very common alternative is volontiers.

For example:

• Paul accepte volontiers.

That also means Paul gladly accepts or Paul is happy to accept.

The difference is mostly stylistic:

• avec plaisir feels very transparent and literal
• volontiers is a common one-word adverb with a similar meaning

Both are natural.

Yes, very often.

On its own, Avec plaisir ! is a common expression meaning:

• with pleasure
• gladly
• you're welcome / happy to

For example:

• — Tu peux m'aider ?
• — Avec plaisir !

So in your sentence, the same expression is simply attached to the verb:

• Paul accepte avec plaisir.