score:8
Where exactly would I be clearing immigration?
Munich. The LH agent must have misunderstood something. The MUC-TXL flight is an in-Schengen flight and there are no facilities to separate people cleared to be in Schengen and those who haven't so you must be cleared first. Your bags go all the way through, you will pick them up in Berlin, maybe that's what the agent told you about.
Given the above answer, is a layover of 1h 20 min enough to make a transfer?
Yes, Munich is one of the most efficient airports, with just 30 minutes minimum connection times, it was specifically built for short connection times and Lufthansa has a hand in running the airport, too (very rare) and so it stays like that. 1:20 is plenty in Munich.
By next flight do they mean the next available flight,
Exactly that. There are so many flights I'd expect at least one per hour but since you are flying in the morning, even more frequent is possible depending on which day.
Note: " or the same flight but the next day? " couldn't work in practice, not every flight is daily, some flights run on a specific day (for example, every Tuesday) or days (for example, every Wednesday and Saturday). Even worse the flight might just never fly again because the schedule changes the next day for example at season boundaries. Also, the rule is there to help the stranded customer not to frustrate them even more, missing a flight is bad enough.