The 404 error shown should not be a reason for concern and isn't the source of the issue. That error is indicating that your browser looked for a favicon.ico (the little icon shown in the title of your page) and did not find one present. This, while minor could be an issue for the template developers since it is related solely to the static assets on your site.
The 503 error is concerning as this error means that the web server was not able to reach the service it was looking for in order to complete the request. This can happen if you are using nginx as your web server with php-fpm if there is an issue with the PHP service.
The first step would be to review your logs where details of the 503 error should be recorded.
If you are using Nginx the log file can be found in:
If you are using Apache on Ubuntu/Debian the log would be:
If using Apache on CentOS/Fedora the log would be:
If you can share what you find there related to this request it would be helpful in figuring out the underlying issue.