Den här möjligheten gör henne glad, och hon hoppas att arbetsgivaren är nöjd med hennes ansökan.

Both are possible, but they differ in style and structure:

• den här möjligheten = very common, spoken/neutral style. It uses a demonstrative phrase (den här) plus the definite noun (möjligheten).
• denna möjlighet = more formal/written. With denna, the noun is usually indefinite: denna möjlighet (not denna möjligheten in standard Swedish).

Because the sentence refers to a specific opportunity already identified by den här (this). In Swedish, when you use den här / den där, the noun typically takes the definite ending:

• den här möjligheten = this opportunity
• en möjlighet = an opportunity (unspecified)

Möjlighet is an en-word (common gender). The definite singular ending for many en-words is -en:

• möjlighet = opportunity (indefinite)
• möjligheten = the opportunity (definite)

Because henne is the object form (her), while hon is the subject form (she).

• hon = she (subject)
• henne = her (object)

Here, Den här möjligheten is the subject, and it affects her, so Swedish uses henne.

The adjective agrees with the person being described. Henne refers to one person, so it’s singular:

• glad = glad (common singular / base form)
• glada = glad (plural) or sometimes polite/plural forms (not relevant here)

So gör henne glad = makes her glad.

It’s optional and style-dependent. Swedish often uses a comma to separate two independent clauses, especially if they are a bit long or clearly separate ideas:

• Den här möjligheten gör henne glad, och hon hoppas ... (slightly more separated) Many writers would also omit it:
• Den här möjligheten gör henne glad och hon hoppas ... (more flowing)

Both can be acceptable depending on style guidelines.

Because after att, you’re in a subordinate clause, and Swedish uses straight word order there:

• subordinate clause: subject + verb → arbetsgivaren är Inversion (är arbetsgivaren) is typical in questions or in main clauses after something is moved to the front, but not after att in a normal statement.

Common patterns are:

• hoppas att + clause = hope that ...
  Example: hon hoppas att arbetsgivaren är nöjd
• hoppas på + noun = hope for ...
  Example: hon hoppas på ett jobb = she hopes for a job

So att is used when what follows is a full clause.

Swedish often uses the definite form when the listener can identify which one is meant from context, even if English might use an in some situations. Here it’s likely the employer connected to her application:

• arbetsgivaren = the employer (the relevant/known one) If it were truly any employer in general, you might see:
• en arbetsgivare = an employer

Arbetsgivaren is singular, so the adjective is singular:

• nöjd = satisfied (singular; common form used for a person)
• nöjda = satisfied (plural)

So arbetsgivaren är nöjd = the employer is satisfied.

Nöjd med is the standard collocation for satisfied with:

• vara nöjd med något = to be satisfied with something
  So: nöjd med hennes ansökan = satisfied with her application

Other prepositions would generally sound wrong here in normal Swedish.

Swedish possessives normally come before the noun:

• hennes ansökan = her application

Putting it after the noun is not the normal possessive pattern (Swedish doesn’t use application hers as a standard structure).

No. Possessive pronouns like hennes don’t change for grammatical gender in Swedish:

• hennes bok (en-word)
• hennes hus (ett-word)
• hennes ansökan (en-word)

It stays hennes regardless.

A few common ones for English speakers:

• möjligheten: the j is a y sound, and the stress is typically on möj- (roughly MÖY-li-he-ten).
• arbetsgivaren: compound word; keep it as one unit with main stress early: AR-bets-gi-va-ren.
• ansökan: ö is not English o; it’s a front rounded vowel (similar to German ö).