These questions revolve around
DH and ECDH vs
DHE and ECDHE. Specifically within the context of TLS/SSL. There are three questions in total (and a fourth bonus question).
The goal of using Diffie-hellman at all in TLS/SSL is to avoid the case where the contents of the certificate are the sole source for seed value for generating symmetric keys. In a non Diffie-Hellman TLS/SSL session (aka, an RSA key exchange), if an attacker where to capture all the traffic, then at some time in the future acquire the private key, they can decrypt the entire content of the previously captured data.
Using Diffie-Hellman, however, a compromised private key won't instantly allow the attacker to decrypt all the prior captured cipher text. Which is why you'd want to use Diffie-Hellman for key exchange, instead of RSA.
But from my reading, DH and ECDH perform the diffie-hellman key exchange using values stored in the certificates. My first question: is specifically which of the DH values are stored in the Public Certificate and Private Key?
Having said that, my second question is does using DH/ECDH provide forward secrecy?
Lastly, it would seem that there is a bigger security increase going from DH/ECDH to their ephemeral counterparts (DHE/ECDHE), than there would be from going from RSA to DH/ECDH. Given that, my third question is what would the benefit be (if any) of using DH/ECDH instead of the ephemeral counter-parts?
bonus question: Does DH/ECDH have much usage in present day SSL/TLS implementations? Or is it mostly just RSA and DHE/ECDHE that are typically used for key exchanges?