There is an API to get the count of posts for a given context. pxGetPulseMessagesCount
But there is no OOTB api to get the count of posts that are shown in the feed of Pulse tab. We can plan to consider this as Feedback.
Also showing the unread posts as new is something which is not available in Pulse posts. but there is concept of Notification Gadget in 7.3 and in that all the Pulse posts are shown as notifications. There the concept of read and unread is available.