When renewing a certificate with a new private key, what happens with browsers that connected previously? Will the old certificate be cached and requests encrypted incorrectly? Is it possible at all to run multiple servers load balanced at layer 4 with some of them having new and others old certificates without causing connections to fail assuming no sticky sessions are used?
Clients usually do not cache SSL/TLS certificates. Only if you use the "Public Key Pinning Extension for HTTP (HPKP)" client do cache and check the provided certificate (or to be exact certain properties of that certificate). For changing the certificate HPKP can "allow" multiple certificates (e.g. one old and one new).
Regarding the load balancer: If they work on osi layer 4 I assume they work on TCP level. Therefore each server behind the balancer establishes it's own SSL/TLS session. If the sessions are not shared among the servers there should not be a problem event if not all servers use the same certificate - as long as all certificates are valid. Clients can provide an SSL/TLS session id when starting the SSL/TLS connection, but the server decides if the session is known or not. Therefore if the client references a session from a different server nothing bad happens, client and server just establish a new session.