Currently, Travis for Jira only supports Jira Cloud instances because there is no comparable feature in Jira Software Server. If the developer panel becomes available in Jira Software Server we will consider adding support for Jira Server as well.
Currently, we do not send Travis CI deployment information to Jira. However, it's surely something we would like to do. The main reason that this feature is currently lacking are some open questions around the mapping between the Travis CI's and Jira's concept of deployment.
Please have a look at the pricing tab on the Intercom for Jira Marketplace listing to get a rough idea. Travis for Jira follows the Atlassian Marketplace pricing model which is priced based on the number of active users in your Jira instance. Make sure you deactivate Jira users that are not in use to make sure you don't pay for them.