So if the folder is listed under Local Folders, it will not sync with the server. This screenshot shows contacts but the same is true with calendars:
When you cerate an item, make sure to save it in the correct folder:
You can also drag items from other folders into the Local Folder calendar or contacts.
Yes, if the contacts have birthdays, they will be displayed in the Agenda.
I gave some instructions for enabling birthdays in another of your posts. One of the options in the Agenda settings is to choose which folders are included.
Make sure you select the appropriate Local Folders where you have saved those contacts.