No quiero que mi hija salte en el sofá.

Because salte is the present subjunctive of saltar.

After querer que when you are talking about what you want or do not want another person to do, Spanish normally uses the subjunctive:

• Quiero que ella salte
• No quiero que mi hija salte

If the subject stays the same, Spanish usually uses the infinitive instead:

• No quiero saltar en el sofá = I do not want to jump on the sofa

So:

• salta = present indicative / command form, not correct here
• saltar = infinitive, used when there is no change of subject
• salte = correct form after quiero que with a different subject

Que introduces the second clause: que mi hija salte en el sofá.

In English, that is often optional:

• I do not want my daughter to jump
• I do not want that my daughter jumps is not natural English

But in Spanish, que is required in this structure:

• No quiero que mi hija salte...

You cannot say:

• No quiero mi hija salte... ✘

So que is the normal link between the main verb of wanting and the action done by someone else.

Because mi hija is not the direct object of quiero here. It is the subject of the subordinate clause:

• No quiero = main clause
• que mi hija salte en el sofá = subordinate clause

So mi hija is the person doing the jumping.

Compare:

• Quiero a mi hija = I love my daughter / I care about my daughter
  Here mi hija is a direct object, so the personal a appears.

• No quiero que mi hija salte = I do not want my daughter to jump
  Here mi hija is the subject of salte, so there is no a.

Spanish often uses en where English uses on for location, especially with furniture and places where someone is positioned.

So:

• en el sofá = on the sofa

This is the most natural everyday wording.

You might also see:

• sobre el sofá

But sobre sounds more like on top of or emphasizes physical position more literally. For normal everyday speech about someone jumping on the couch, en el sofá is the usual choice.

Because the subjunctive is triggered in the subordinate clause, not automatically in the main clause.

The main clause is:

• No quiero = I do not want

That is a straightforward statement about the speaker’s real feeling, so quiero stays in the indicative.

The wish or desired action is in the second clause:

• que mi hija salte

That is where Spanish uses the subjunctive, because it is something wanted, not simply presented as a fact.

So the pattern is:

• [main clause in indicative] + que + [subjunctive]

Yes, if it is already clear from the context who you are talking about.

Spanish often leaves out subjects when they are understood. So:

• No quiero que salte en el sofá

can mean I do not want her to jump on the sofa, if everyone already knows who she is.

But mi hija is included here to make the subject explicit and avoid any confusion.

In modern standard Spanish, possessives like mi, tu, su, nuestro, etc. normally go directly before the noun:

• mi hija
• tu casa
• su coche

So la mi hija is not standard modern Spanish in this context.

Also, with close family members, Spanish very commonly uses the possessive exactly like this:

• mi madre
• mi padre
• mi hijo
• mi hija

Yes. That is grammatical, but it gives a slightly different nuance.

• No quiero que mi hija salte en el sofá
  This is the more general or neutral version: I do not want my daughter to jump on the sofa.

• No quiero que mi hija esté saltando en el sofá
  This emphasizes the action as ongoing or in progress, more like: I do not want my daughter to be jumping on the sofa.

So both are possible, but the original sentence is simpler and more natural in many everyday situations.

Yes, sofá is a masculine noun:

• el sofá
• los sofás

The accent mark shows that the stress falls on the last syllable: so-FÁ.

It keeps the accent in the plural too:

• sofá
• sofás

So in the sentence, el sofá is correct because the noun is masculine singular.