I wish I had seen this list of names that he knows:

http://www.gameranx.com/features/id/31547/article/fallout-4-list-of-names-codsworth-says/

I intentionally picked Fred (a name I don't like), just becuase I thought it would be on the list.

1) Is there a way to change my name (out of the Vault)?

2) Will Codsworth say my (new) name in future conversations if I changed it to one on the list?

1) Yes there is, if you are on the PC as a platform that is. You can use a console command to bring up the new equivalant of the showracemenu.

Quote from Reddit -

It's ShowLooksMenu. You have to enter it a certain way.

ShowLooksMenu Player 1 (for changing hair) or 2 (for plastic surgery)

Make sure you're sitting down when you do it, otherwise the camera will bug out and you won't be able to fix it. So save before using any of the commands. You'll have to be in third person in order to see what you're doing.

2) That one i do not know. I haven't changed name yet, but for me he sometimes called my character mum and sometimes the name. It might be bugged, not sure.

Seems really strange that Fred would not be on the list...

You can find the list on the wiki already. If the name is on the list he'll say it. My characters first name isnt on the list, but his last name is and Codsworth calls me by name.