In short, there's no way to get Lending Library on any non-Kindle device (Kindle app on Nexus 7 will not work for lending benefits). Only Kindle devices, including Kindle Fire tablet support lending (This is Amazon's restriction - their way of pushing their own devices). There is no known workaround for this.
Amazon Instant Video app, which streams videos, including Prime selection is only available on Kindle Fire and Apple iPad (again, Amazon doesn't want to make it available for other Android devices like Nexus, that compete with Kindle).
You can browse Instant Video Selection via Android browser on Nexus 7 - any browser EXCEPT Chrome will work, and will stream video as long as Adobe Flash app is installed. Chrome has declared they will not support Flash moving forward, in preference to HTML5, but Amazon VOD requires Flash. So in short, install any free Android browser, and install Flash, then browse via:
You can download the App store via going to the Amazon app website, inputting your email, opening the email you get from Amazon on your Nexus and then following the directions from there. That itself won't get you prime benefits.
And looks like all you really need is to install flash and get a flash capable browser (firefox or Xscope) to get Amazon Prime, movies and media.