Tweak/add feature to Wordpress

I have a Wordpress theme that I am working on - I want to add the following functionality/tweak (this can be through plugins or self-coded, as long as it works):

--> Only show a set number of posts on the home page - when non-registered users click on the image of the post they will get a pop-up asking them to either Login or Sign up using Facebook or email&password etc.

(Please see images attached to give you an idea)
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If this goes well, I have several more features that I will pass on to you.

Thanks