is there any attribute in JMX dump or function that can be started via JMX client that can tell me that Share is up and running or at least it's active?
Alfresco Share does not provide any specific JMX beans / attributes to check. You may use JMX attributes provided by your application server (e.g. Tomcat) to check if the /share web application has been successfully deployed. But that of course does not provide you with the information that the web application has in fact been initialised correctly (only that it was deployed without an error).
In addition, just to say Share rarely is deployed incorrectly and problems are more commonly related to Alfresco repository. So maybe a direct monitoring of Alfresco share url is enough. Relating to Axel's comment Catalina/Manager/Share JMX object for example is able to show Share active sessions.
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