Ok - the next thing is to repeat what you did before but this time use the Network tab in the developer tools. Look for the items that produce the '403' error code. Double click and check the details of those. I doubt you'll find anything more interesting than a 403 was generated.
I think I'm right in saying that Moodle doesn't generate 403 codes, so the web server is doing this. Go back to your host's support and ask them to check the logs at their end (give them your IP address and time to make their lives easier) to establish what generated the 403 errors. Make sure you're absolutely clear that your concern is the 403 errors and not the stuff about the menus and so on (which is a consequence of the 403 errors and not their concern).
I'm prepared to be wrong but I'd be very surprised if this is anything to do with Moodle per se.