AFAIK, PushBullet doesn't allow you to answer phone calls.
The closest analogue it comes to my mind is an early demo of Ubuntu for Android in which they showed the ability to reply to SMS messages from within the Unity desktop (although I'm certain I saw a phone call demo too, just can't find a link to the video). However, in that case, both the Ubuntu desktop and Android were running on the same device, so it was easier (still a notable feat for the time).
This kind of integration between different devices is far more impressive, though.
Nope it doesn't - that's an example of where this is much smoother. FWIW MightyText already does SMS (via an app, not proximity) on Android/Chrome - although it's not Apple smooth in terms of replying either.
I think the right way to go would be syncing the actions as well as data of Android's rich notifications - things like reply to email/archive from chrome notifications, reply to SMS, hangouts.. It'd be Google's universal apps between Android and Chrome - syncing actions. It'd need a lot of developer support but it'd be fantastic if they pulled it off as well as Apple and supported third parties in a serious way.
The closest analogue it comes to my mind is an early demo of Ubuntu for Android in which they showed the ability to reply to SMS messages from within the Unity desktop (although I'm certain I saw a phone call demo too, just can't find a link to the video). However, in that case, both the Ubuntu desktop and Android were running on the same device, so it was easier (still a notable feat for the time).
This kind of integration between different devices is far more impressive, though.