1) HTTP features the "Accept" header which allows the user-agent to specify the format of the response. The "*.json" thing is not standard and breaks HTTP.
2) "tickets/search" endpoint is not a resource. It's a different projection of the "tickets" collection. It's basic filtering. Rule of thumb is whenever you're about to use a verb to name an API endpoint, you can guess you're probably about to do something wrong.
3) A RESTful API needs hyperlinks. Resources must point to each other to help the developer navigate the API tree and access every single point it contains. The API origin (company.supportbee.com) should be the only information I need to discover the structure of your API.
The WWW is an example to follow. It's the most RESTful "service" out there.