När vi ses på fredag, kommer hon att ha skrivit under kontraktet om allt går bra.

Yes. Swedish very often uses the present tense after time expressions to talk about the future, especially for planned/scheduled situations. So När vi ses på fredag literally uses present (ses) but refers to a future meeting because på fredag sets the time in the future. Adding kommer att inside the när-clause would usually feel unnecessary or heavy.

Ses is the -s form of se and in this meaning it’s a reciprocal/middle-voice type: vi ses = we see each other / we meet. It’s extremely common in everyday Swedish for arranging meetings.
Vi träffas also works and is close in meaning, but vi ses is often the default for appointments (and can sound a bit more casual).

Because Swedish has the V2 rule in main clauses: the finite verb is in the second position. The sentence begins with a subordinate clause (När vi ses på fredag), so the main clause that follows starts with the finite verb kommer, and the subject hon comes after it:

• När vi ses på fredag, kommer hon ...
  If you start with the subject instead, you get the normal order:
• Hon kommer ... när vi ses på fredag.

It’s common and helpful, but not always strictly required. Swedish often uses a comma after a fronted subordinate clause to make the structure easier to read:

• När vi ses på fredag, kommer hon ...
  In shorter sentences, people sometimes omit it, but including it is generally safe and clear.

It corresponds to future perfect in English (will have done). Swedish builds it like this:

• kommer ... att (future marker + infinitive marker)
• ha (infinitive of have)
• skrivit (supine form used with ha) So kommer hon att ha skrivit = she will have written/signed (by that time).

Because with ha Swedish uses the supine form, not the participle:

• infinitive: skriva
• past tense: skrev
• supine: skrivit (used after har/hade/ha)
• past participle (adjective-like): skriven/skrivet/skrivna
  In ha skrivit, you need the supine skrivit.

It’s a particle verb: skriva under = to sign (a document). Here under is a verb particle, not a normal preposition meaning under. The object is kontraktet.
You’ll also see it split in some forms:

• Hon skriver under kontraktet.
• Hon skrev under kontraktet.
• Hon har skrivit under kontraktet.

Swedish uses the definite form when the context implies a specific, known item (here: that particular contract). Kontraktet means the contract (the one we’re talking about). Ett kontrakt would introduce it as an unspecified contract.

Often, yes—especially in informal Swedish. Both can be heard:

• more standard/neutral: kommer hon att ha skrivit under
• more spoken: kommer hon ha skrivit under
  Including att is a safe choice for learners and in writing.

Om allt går bra is a conditional subordinate clause (if everything goes well) that adds a condition to the main clause. Placing it at the end is very natural. It can also be moved for emphasis:

• Om allt går bra kommer hon att ha skrivit under kontraktet när vi ses på fredag.
  If you front it, you again get V2 inversion in the main clause (kommer hon), just like in the original sentence.