알겠어요 is better than 알았어(요). 알았어(요) is more like "I see" rather than "I understood". :-)

네. Yes. 알았어요, 알겠어요. I got it, I see. 이해했어요. I understand.

al-ass-eo-yo (for polite informal) al-gess-eup-ni-da (for polite formal)

simplest would be: 1) ne 네 (yes) 2) arasso 알았어 (i get it, i understand) make sure your voice doesn't inclinate upwards, as this can represent a question, instead of an answer.