Can I create a JIRA ticket from a Salesforce custom object?

[#1]

Yes you can! You only need to change the "Case" to your custom object API name. Do note that custom object API name should have the "__c" behind the name.

e.g., http://localhost:6250/plugins/servlet/customware/connector/issue/3/Defect__c/create.action?id={! Defect__c.Id}