1.15.8 is going to reach end of life very shortly. once 1.17 is released. You might want to consider upgrading.
PHP 7.4 shouldn't have any problems with osTicket 1.15.8 to my knowledge.
You did not tell us what webserver you are running.
Environment details? (see Admin panel -> Dashboard -> Information)
The web server does not appear to be handling AJAX calls. You will likely want to call your web host and have them look at the logs for your site to find out why. They are likely blocking something that they shouldn't be with mod_security or something similar. If they cannot or will not change their settings for your site then you would probably want to look for a new web host, or set your own inhouse server.