Let's imagine for a while that Let's Encrypt is an NSA project, and let's imagine that soon, almost everyone will be using it. Would the NSA have any cryptographic advantage because of that, when analyzing SSL/TLS communication?
Very short answer:
Would NSA have any cryptographic advantage because of that,
Yes. And that applies to any certificate authority: Whomever your users trust to authenticate your website is able to forge a valid certificate. Verisign, Let's Encrypt, doesn't matter. They have everything they need to make your user complacently be subject to a man-in-the-middle attack.
Note that it is known that some CAs that are trusted by default on many systems/browsers/clients are subject to laws requiring access to the signing keys. And as far as I remember, there's also already publicized cases where law enforcement used that.
when analyzing SSL/TLS communication?
Aside from the MITM scenario explained above, in theory, no. A CA just signs your public key, it doesn't get any more insight into the encrypted streams.
NO (but maybe yes).
Short answer is sound no, because your web site private key is never sent to letsencypt, and thus encrypted communication between clients and server cannot be decrypted any easier that if you were using any other certificate authority.
The worst NSA could do is same they could do with other CAs (and more easily too with other CAs - because letsencrpyt provides transparency report for all certs issued), and that is force CA to issue fake certificate for your website so they can perform MiTM attack. But that would be bad PR for them.
Or they could of course know protocol weaknessess or TLS implementation bugs, and exploit that -- but that would again work for any other CA as well.
However, it might be possible in general case that letsencypt is more vulnerable - as for letsencypt to work (as opposed to other CAs) you have to use the client software to generate certificates (every 3 months at least) and that software could have (intentionally or not) hidden bugs which might theoretically allow NSA (or other parties) to break into your server and retrieve your private keys (and other data) at which point it is game over. Especially as most of clients need elevated privileges (and official one runs as root IIRC).
Note however there are dozens of letsencypt clients, and it is not even too big problem to write your own, so you can choose one in which security you trust.
No as let's encrypt signs the certificate with its private key which resides on their server.
The private/public key pair is generated and the certificate is signed the private key is never sent to let's encrypt servers but only public key and certificate information are sent.
NSA also uses let's encrypt to issue SSL certificates.