1.Does AV of 12.1.1 support Windows Server 2012?
No, Server 2012 and windows 8 are supported only by SEP 12.1 RU2
http://www.symantec.com/docs/TECH174348
2.Can the new version of AV 12.1.2 work on SEPM (management console) version 12.1.1.?
Yes, it will work. Although recommended would be to have the SEPM in the same or higher version than its clients.
3.Can I upgrade SEPM from V12.1.1 to AV 12.1.2 without failure or loosing client connections currently working?
Yes, it should not create any issues- just follow the migration documentation:
http://www.symantec.com/docs/TECH197426
4.What is the next version of Live Update Administrator? Are there some steps I can use to upgrade?
The latest LUA version is 2.3.2
https://www-secure.symantec.com/connect/blogs/liveupdate-administrator-lua-version-232-now-available