Even with the pre-installed google contacts app, you could add contacts locally. If you don't trust a google app, you could use an AOSP contacts app if you have an AOSP rom.

- Generally most flagships today have an always on screen, but it's still not a native feature on AOSP android (at least I don't think so). There are some phones that have a glance feature, that shows the AOD but it's not always on, only activated if you pick up the phone. Having an OLED screen would probably help with battery, so that just depends on what type of phone you have.
- I don't think there are any social media feeds on Android, but there are apps that combine multiple apps such as twitter and facebook into one app, but those are just html wrappers with a few modifications.
- The apps are generally good looking in the google sense (material design), but it's not like N9's apps.
- I'm not sure if the Gmail app stores emails locally, but you can access mails online if you've opened it recently (probably because of cache)

If you are hoping to match how N9 works, Sailfish OS is the most similar since the same guys are working on it. It has a social feed, and has a similar design like the N9 (in addition to silica UI).

Let me know if you need anything else