No. An iPhone can be synced with only one iTunes library at a time. Erasing and syncing replaces the contents of this iPhone with the contents of this iTunes library.
Basically you'd want to get the music to the library first, then sync it to the phone after you get it set up in your library. iTunes will not transfer songs off the phone that were not purchased through the iTunes store, so they need to be in your library beforehand. You might consult with the googles if there are other third party ways to do that.
The 5 compy authorization is for iTunes account authorization, which means transfering music bought through itunes, apps, etc.
The computer authorizations are for the number of different computers that can access your iTunes account. If you use someone else's computer with their iTunes account or library, it will give you that message. If you still have room for more computers to authorize on your account, you can log on a different computer, authorize it and sync your music.