“to follow up formally”
Follow up means to pursue the next steps after an earlier contact, complaint, or inquiry. Formally means through an official process. The manager needs a written record of dates and times, rather than only a spoken complaint, before proceeding officially.
“He couldn't have been nicer about it”
Couldn't have been plus a comparative expresses an emphatic positive: it could not have been any more so. Compare It couldn't have gone better. Although grammatically negative, the phrase praises. Tina says her neighbor responded as pleasantly as possible, even though the noise continued.
“We escalate.”
Escalate generally means to increase or intensify. In complaints and customer support, it often means to move an issue to a higher level or authority. Here, if the warning fails, the record goes to the owner to consider whether the lease has been violated.