Lots of good answers here. I'll integrate some of the answers I think are most relevant and add some more suggestions.
1) Should JWT token validation be limited to verifying the signature of the token itself, relying on the integrity of the server secret alone, or accompanied by a separate validation mechanism?
No, because of reasons unrelated to the compromise of a token secret. Each time a user logs in via a username and password, the authorization server should store either the token that was generated, or metadata about the token that was generated. Think of this metadata as an authorization record. A given user and application pair should only have one valid token, or authorization, at any given time. Useful metadata is the user id associated with the access token, the app id, and the time when the access token was issued (which allows for the revocation of existing access tokens and the issuing of a new access token). On every API request, validate that the token contains the proper metadata. You need to persist information about when each access tokens was issued, so that a user can revoke existing access tokens if their account credentials are compromised, and log in again and start using a new access token. That will update the database with the time when the access token was issued (the authorization time created). On every API request, check that the issue time of the access token is after the authorization time created.
Other security measures included not logging JWTs and requiring a secure signing algorithm like SHA256.
2) If JWT signature verification is the only means of validating tokens, meaning the integrity of the server secret is the breaking point, how should server secrets be managed?
The compromise of server secrets would allow an attacker to issue access tokens for any user, and storing access token data in step 1 would not necessarily prevent the server from accepting those access tokens. For example, say that a user has been issued an access token, and then later on, an attacker generates an access token for that user. The authorization time of the access token would be valid.
Like Akshay Dhalwala says, if your server-side secret is compromised, then you have bigger problems to deal with because that means that an attacker has compromised your internal network, your source code repository, or both.
However, a system to mitigate the damage of a compromised server secret and avoid storing secrets in source code involves token secret rotation using a coordination service like https://zookeeper.apache.org. Use a cron job to generate an app secret every few hours or so (however long your access tokens are valid for), and push the updated secret to Zookeeper. In each application server that needs to know the token secret, configure a ZK client that is updated whenever the ZK node value changes. Store a primary and a secondary secret, and each time the token secret is changed, set the new token secret to the primary and the old token secret to the secondary. That way, existing valid tokens will still be valid because they will be validated against the secondary secret. By the time the secondary secret is replaced with the old primary secret, all of the access tokens issued with the secondary secret would be expired anyways.