Yosemite changes QWERTY keyboard layout
As far as I know, Yosemite Romaji IM's failure to honor option and shift dead keys is a bug that Apple will have to fix via an OS update.  It has been reported elsewhere (e.g. https://discussions.apple.com/message/26864874#26864874  ).  It would be good to use Apple's feedback channel to encourage quick action on this.

The OS X 10.10.1 update does not fix this bug in Kotoeri, but the 10.10.2 update of Jan 27, 2015 DOES fix it.

The behavior described for the US International PC keyboard layout is not a bug, but intended.  People who use this keyboard want the ` ~ ^ " ' keys to be dead keys for producing accented characters, and they learn to add a space when they need the stand alone version.

I have not seen any other reports of the US keyboard layout not working right in Yosemite.  When the US flag is showing at the top right of the screen, you should be able to use the standard option dead keys to produce accented characters without problems.  You should also be able to use the Character Picker, where you hold down the key for the base letter and a popup menu appears where you can select accented versions.  If that is not the case for you, the OS is broken somewhere and may need to be reinstalled.