From what I could find, the lack of forward secrecy is due to the fact that multiple sessions use the same key.
This is not about using the same key inside multiple sessions.
All ciphers using the RSA key exchange lack Forward Secrecy since an attacker could passively sniff the traffic and later decrypt the sniffed traffic if the attacker gets access to the servers private key. This is not possible with ciphers using Diffie-Hellmann (DHE, ECDHE), i.e. these ciphers provide Forward Secrecy.
Or as written at Wikipedia:
A public-key system has the property of forward secrecy if it generates one random secret key per session to complete a key agreement, without using a deterministic algorithm.
With RSA key exchange there is a deterministic algorithm based on the servers private key, with DH there is no such determinism.
How/where are the keys persisted and for how long?
Forward Secrecy (or lack of) is not about the persistence of the keys but about the ability to recreate the keys in a deterministic way, in case of RSA key exchange based on the private key of the certificate.
Can this compromise sessions which will happen in a relatively far future, say half a year from now?
This is about a "compromise" of a session but that the encryption keys can be recreated if the session was sniffed and later the private key gets known. This can be years after the session got sniffed.