Son beau-père apporte une bouteille de jus, et sa belle-mère prépare un dessert simple.

Because in French, possessive adjectives like son, sa, and ses agree with the gender and number of the noun being possessed, not with the gender of the owner.

• beau-père is masculine, so French uses son
• belle-mère is feminine, so French uses sa

So both can mean his or her depending on context:

• son beau-père = his stepfather / father-in-law or her stepfather / father-in-law
• sa belle-mère = his stepmother / mother-in-law or her stepmother / mother-in-law

French does not usually distinguish his and her in these possessives the way English does.

It can mean either one, depending on context.

• beau-père can mean stepfather
• beau-père can also mean father-in-law

Likewise:

• belle-mère can mean stepmother
• belle-mère can also mean mother-in-law

French uses the same words for both ideas, so you usually rely on the situation to understand which meaning is intended.

These are fixed compound nouns in French, so they are normally written with a hyphen:

• beau-père
• belle-mère

Many family relationship terms in French are written this way. The hyphen shows that the two parts work together as one unit.

Historically, beau does mean beautiful or handsome, and belle is its feminine form. But in words like beau-père and belle-mère, you should usually learn them as complete expressions, not as literal combinations.

So it is better to think of:

• beau-père = a family relationship term
• belle-mère = a family relationship term

rather than trying to translate them word by word as beautiful father or beautiful mother.

Because after a container word like bouteille, French normally uses de to show what the container holds.

So:

• une bouteille de jus = a bottle of juice
• un verre d’eau = a glass of water
• une tasse de café = a cup of coffee

Using du here would sound wrong in standard French. The structure is:

container + de + contents

French usually requires an article where English sometimes does too, and sometimes not. Here, both nouns are singular countable nouns, so French uses the indefinite article:

• une bouteille = a bottle
• un dessert = a dessert

This is very normal in French. You generally cannot just say apporte bouteille or prépare dessert.

In French, most adjectives come after the noun. Since simple is one of those adjectives, French says:

• un dessert simple

not usually:

• un simple dessert

However, moving an adjective before the noun can sometimes change the meaning or emphasis. For example:

• un dessert simple = a dessert that is simple
• un simple dessert = merely a dessert / just a dessert

So in your sentence, simple comes after the noun because it is describing the dessert in the ordinary way.

Because both verbs are conjugated in the present tense for il / elle / on.

The subjects are:

• Son beau-père = he
• sa belle-mère = she

So the verbs are:

• apporter → il apporte
• préparer → elle prépare

That is why you see:

• Son beau-père apporte...
• sa belle-mère prépare...

Because French, like English, can sometimes omit the repeated subject when a new noun phrase already gives the subject clearly.

Here the second subject is explicitly stated as:

• sa belle-mère

So the structure is completely natural:

• Son beau-père apporte..., et sa belle-mère prépare...

French only needs one verb per subject here, and the sentence is balanced and clear.

You usually know from context.

In French:

• son can mean his, her, or its before a masculine singular noun
• sa can mean his, her, or its before a feminine singular noun
• ses can mean his, her, or its before a plural noun

So in this sentence:

• son beau-père could be his stepfather/father-in-law or her stepfather/father-in-law
• sa belle-mère could be his stepmother/mother-in-law or her stepmother/mother-in-law

French leaves that detail to context more often than English does.

Yes, a few useful points:

• son sounds roughly like so(n) with a nasal vowel; the n is not fully pronounced like in English
• beau sounds like bo
• père sounds like pair
• belle sounds like bell
• mère sounds somewhat like mair
• jus sounds like zhu, with the j like the s in measure
• dessert ends with a pronounced t sound in French: day-sair roughly
• simple is roughly sanh-pl, with a nasal vowel at the start

Also, in careful speech, et sa belle-mère is usually pronounced smoothly, but et does not create a strong liaison here.

Yes. The French present tense often covers both ideas, depending on context.

So apporte can mean:

• brings
• is bringing
• sometimes even does bring, depending on emphasis

And prépare can mean:

• prepares
• is preparing

In a simple standalone sentence like this, English often translates them as brings and prepares.