How to I add translations for Month and Day in a date field?
You can translate everything in Administration » Configuration » Regional and language » Translate interface (admin/config/regional/translate/translate)

Concerning dates, you usually need to translate the following strings :


!month-name |January|February|March|April|May|June|July|August|September|October|November|December : you must keep !month-name and | in the translated string
every string starting by !long-month-name
!day-name Sunday|Monday|Tuesday|Wednesday|Thursday|Friday|Saturday 


And you should be good.

Even better, if you know several people interested in maintaining a language translation, you should go to https://www.drupal.org/node/807756 and reopen the issue. The goal is to have one place to improve and maintain the translation in https://localize.drupal.org/