Terminal with real time dmesg output

You're going to need to edit your /etc/syslog.conf.

Adding a line like:

kern.*          /dev/tty10

Would output all kernel info to that tty. You'll need to figure out which tty is assigned to contol-alt-f12. There are many more examples in man syslog.conf.

You need configure this in /etc/syslog.conf (or /etc/syslog-ng.conf depending on syslog version) directing kernel messages to /dev/vc/12 (for alt-ctrl-12).

If there is a log file already, a manual solution is to do something like:

tail -f /var/log/kernel.log > /dev/vc/12

But the permanent way is to set it up for your flavor of syslog

(in my archlinux install there is a predefined rule for this in /etc/syslog-ng.conf, uncommenting it would turn exactly this feature on)

Is this not what xconsole does in X11?
Otherwise you tail -f /dev/console. If I'm wrong here, and dmesg shows more than the console, then you need to look into syslogd.
Set your /etc/syslog.conf to either send kern messages to console, or to whichever stty is attached to the alt-ctrl-F12, which should be /dev/vc/12.
So that's /etc/syslog.conf:

kern.*  /dev/vc/12

You can simply run

cat /proc/kmsg

for realtime dmesg output in current terminal