Dear @LLukas
Could we please make sure any third-party plugins are de-activated? do you have any installed currently?
Do you know what PHP version your website is running?
What version of Lay Theme are you installing?
This thread has a very similar issue. The user had to contact the Webhost regarding mod_security or PHP issue:
http://laythemeforum.com:4567/topic/5752/http-error-500-after-project-update/6
Talk soon @LLukas & best wishes
Richard