The BR file of births is the one to use and includes living and dead children. The KR file only has living children.
Kaplan-Meier survival analysis gives a survival curve taking into account right-censoring (i.e. loss to follow-up). We have that if, for example, we want under-five mortality rates for the most recent five-year period since not all children will have been 5 years of age or older at the time of the survey: their experience is truncated by the interview and we don't know what will happen to them at ages less than 5 years in the future. Note that the experience of the older children under age 5 is being used to estimate mortality at the older ages. They will have been born earlier but died recently. There may be a problem with sample size for these estimates. For the mortality at younger ages, say under age 1 year, the experience of most of the children born in the last five years is used.

Kaplan-Meier is usually used to compare two or three groups (e.g. control versus treatment, urban vs. rural, male vs. female).

The Cox proportional hazard regression procedure estimates the hazard rate of a group with a selected characteristic compared with a baseline and is generally used with multiple correlated variables to find the effect of each.