Support

Regarding the emergency: You are using CloudFlare. You can block these IPs, their ASNs, or all of Russia on CloudFlare's end. You could even set up a rule to allow only specific IPs to access /administrator/* URLs on your site. This is why my recommendation has always been to have one of these services in front of your site; you can cut a lot of malicious traffic with the efficiency (and grace) of a sledgehammer.

Now, about the issue.

First, check your mail settings. Can your server send email? How long does sending an email take? How many Super Users do you have? Whenever you save Admin Tools' configuration an email is sent to all Super Users with troubleshooting information, in case they lose access to the site. If your site hangs trying to send these emails, or takes too long, you will have a timeout.

If that is not the case, there is a very good chance that your host has its own server protection –most likely Apache's mod_security2 with the OWASP core ruleset– which gets in the way of saving the Admin Tools configuration options. Contacting your host is the only way to check that, and they will be able to tell you what code to add to your .htaccess file to disable the security rule which causes the problem. Moreover, since you have CloudFlare which implements the very same security rules, you can also ask your host to disable that security for your site.