Latest has no code to limit to recent 3 months. It simply queries database and pulls all the groups based off the limit you specified in the module ordered by creation date. Next release will feature a completely rewritten module, but again there's no reason for it to be doing that and could not duplicate your issue. Popularity count is based off how many users belong to that group. Groups can stop showing if display access to their category or the groups themselves is lost. For example if you unpublish their category then you'll lose display of the groups.

This is a bug with the way group and category access is handled in current releases of GJ. What is happening is it's querying the database for the latest 5 groups for example. Then it checks for access to those groups and removes them as necessary. Unfortunately this includes unpublished, etc.. groups so say you've 5 unpublished groups but have 3 published groups then those 3 won't be shown as it grabbed the first 5 and removed them due to being unpublished.

This has been fixed for next release as the entire process of obtaining groups and categories is completely rewritten. A workaround would be to increase the display limit higher or publish those unpublished groups or unpublished categories that contain groups.

At any rate it's a bug and is fixed for GJ 2.4. I apologize for the inconvenience.