Hon vill ansöka om ett nytt jobb, men hennes ansökan är inte helt färdig.

Because vill is a modal verb. After Swedish modal verbs (like vill, kan, måste, ska) you normally use the infinitive form of the next verb, not a conjugated form:

• Hon vill ansöka = She wants to apply
  Not hon vill ansöker.

In Swedish, the usual pattern is ansöka om + something when you apply for something:

• ansöka om ett jobb = apply for a job
  You can also use ansöka till in some contexts (often institutions/programs), e.g. ansöka till universitetet (apply to the university), but for jobs, ansöka om is the standard choice.

Because jobb is an ett-word (neuter): ett jobb.
Adjectives agree with gender in the indefinite singular:

• en-word: en ny bil
• ett-word: ett nytt jobb (ny → nytt)

Both can translate as job/work, but:

• ett jobb usually means a specific position/employment (a job you can apply for)
• ett arbete can mean work in a broader sense, and can also mean a job, but often feels more formal or general
  So ansöka om ett nytt jobb is the most natural for “apply for a new job (position).”

The noun is en ansökan (an application).
When a noun is used with a possessive like hennes (her), Swedish typically uses the definite form of the noun:

• en ansökan = an application
• ansökan = the application
• hennes ansökan = her application (literally: her-the-application)

You usually do not add an article (en) when you have a possessive (hennes).

Sin/sitt/sina is a reflexive possessive that refers back to the subject of the same clause.

In the second clause, the subject is hennes ansökan (her application), not hon. So you can’t use sin there; you use hennes to mean “belonging to her (someone else than the clause subject).”

If the clause subject were hon, then sin would be possible:

• ... men hon har inte gjort sin ansökan färdig. = ...but she hasn’t made her application finished.

In a main clause, Swedish normally places inte after the finite verb:

• hennes ansökan är inte helt färdig
  This is the typical pattern: Subject + Verb + inte
  • rest.

But in a subordinate clause, inte comes before the finite verb:

• ... för att hennes ansökan inte är helt färdig = because her application isn’t completely finished

Predicate adjectives (after är, blir, etc.) agree with the subject:

• en ansökan (common gender, singular) → färdig
• ett brev (neuter, singular) → färdigt
• plural → färdiga

So ansökan är ... färdig is correct because ansökan is an en-word singular.

helt is an adverb meaning completely/entirely.
So inte helt färdig means not completely finished (partly done, but not fully ready).

Without helt, inte färdig is a bit stronger/more general: not finished/not ready.

Often, yes:

• ansökan är inte helt klar can sound very natural too.

A rough nuance:

• klar = ready/done (often about being ready to submit/use)
• färdig = finished/complete (often about the work being completed)

In many everyday situations they overlap, and both can work here.

Not strictly. Swedish comma rules are more flexible than in English. With two main clauses joined by men (but), many writers include a comma for clarity:

• Hon vill ansöka om ett nytt jobb, men ... You’ll also see it without a comma, especially in shorter sentences:
• Hon vill ansöka om ett nytt jobb men ...