Questions about MyBB Backdoors

Well there is no way to "check for backdoors" as backdoors are found by hackers. There isn't like a tool where you can run and will tell you all the backdoors and vulnerabilities on your website. You could however deploy a Web Application Firewall or scan your website for vulnerabilties and patch it. By outdated theme I would presume that this theme hasn't been updated for some time. To my understanding of MyBB forums, I think the structure of themes have no changed much so it shouldn't really effect you.

Good luck