Learn Italian: Finito il pranzo, ero così sbadato che ho rovesciato il caffè. - Having finished lunch, I was so careless that I spilled the coffee.

Unlike phrases such as in modo che (so that) which express an intended purpose and require the subjunctive, così ... che expresses an actual, factual result. Because spilling the coffee really happened as a consequence of being careless, we use the normal indicative mood (ho rovesciato).

Could I use talmente instead of così in this sentence?

Yes, così and talmente are perfectly interchangeable when modifying an adjective like sbadato. You could say ero talmente sbadato che and the meaning remains exactly the same.

How does Finito il pranzo mean "Having finished lunch" without using a helper verb like "avere"?

This is a common Italian shortcut called an absolute past participle. You can drop the helper verb and just use the past participle (finito) to mean "after something was done." It is a more elegant and concise way to say Dopo aver finito il pranzo.

Why do we use the imperfect tense for ero but the passato prossimo for ho rovesciato?

The imperfect (ero sbadato) describes your ongoing state or background condition at the time. The passato prossimo (ho rovesciato) is used for the specific, single completed event that happened as a sudden result of that ongoing state.