Learn Swedish: Orkestern kommer att ha spelat symfonin innan klockan åtta. - The orchestra will have played the symphony before eight o'clock.

It is built just like in English: the future marker kommer att (will), plus the infinitive ha (have), plus the supine form spelat (played). In everyday speech and informal writing, the att is very often dropped, making it kommer ha spelat.

The lesson says we can also use ska ha. Would ska ha spelat mean the exact same thing here?

Not exactly. Kommer att ha acts as a neutral prediction about the future (we predict they will have finished it by then). If you say ska ha spelat, it implies intention, a schedule, or an expectation, meaning they are supposed to have played it before eight according to the concert program.

Wait, isn't före the preposition for "before" a time, while innan is only used before a whole clause?

Traditionally, yes! You might have been taught to use före klockan åtta (before eight) and innan vi äter (before we eat). However, in modern Swedish, innan has crossed over and is completely accepted as a preposition of time. Hearing innan klockan åtta is extremely common today, so you don't need to strictly separate them anymore.

Does the verb form spelat ever change depending on what the orchestra is playing?