TLS/SSL's usage of Non-Ephemeral DH vs DHE
A Diffie-Hellman key agreement has the following general form, presuming it is done in a group $G$ of order $q$ with generator $g$:


$A$: Generate $x \in \mathbb{Z}_q$ at random. Calculate $X = g^x$
$B$: Generate $y \in \mathbb{Z}_q$ at random. Calculate $Y = g^y$
$A \to B$: $X$
$B \to A$: $Y$
$A$: Calculate $S = Y^x$
$B$: Calculate $S = X^y$


First question: The certificate of $A$ contains the public key $X$ and a reference to the group $G$. The group might either be explicitly specified (as is common in the case of DH) or referenced by an identifier (as is common in the case of ECDH). The corresponding private key is the secret exponent $x$. Note that such certificates are rare; most commercial certificates contain RSA public key and might not be used for DH or ECDH (obviously).

Second question: No. If static-DH/ECDH or ephemeral-static-DH/ECDH is used, at least one party will always use the same private key. If that private key is apprehended by an adversary, all prior shared secrets that involve the key pair that private key belongs to, might be calculated from the protocol messages.

Third question: The security of RSA key transport trivially requires that the client (who generates and sends the RSA encrypted pre master secret) has access to a good (pseudo) random number generator. This is true for all attack scenarios; if the client random generator is compromised, RSA key transport will provide no secrecy. For DH/ECDH on the other hand, there are attack scenarios where the client doesn't even need a random generator.

Fourth question: AFAIK DH/ECDH are rare for SSL/TLS, because they require that the server has a DH/EC certificate, and those are rare.