Yes, you can safely probe a tube to find the heater. Everything else in a tube is just some metal with vaccuum between it and the other parts. The small voltages and currents of a ohmmeter won't damage anything, not even close.
The "12" in the part number indicates that the heater is intended to be powered from 12 V. This one has a separate heater, which usually means it was designed to be run directly from a 12 V tap of the main power transformer. The heater is electrically isolated from the cathode so that the AC doesn't get into the signal (at least mostly not).
Here is the section on this tube from the 1960 RCA Receiving Tube Manual:
This confirms the heater is intended for AC and also tells you the pinout explicitly. Keep in mind that tube pins are numbered clockwise when looking at the tube from the bottom, because that's how you'd be looking at it when soldering wires to the socket. The convention is that there is always one place around the circle that is special, usually a gap in the pins, but it could also be a mark or a nub sticking out or something else. You start with pin 1 going clockwise from that specially marked place. In this tube, the special place is marked by a larger gap between the pins than the other gaps. That is very common.
Just because the heater can be run from AC, doesn't mean it needs to be. You can safely run this heater from a 12 V battery like a car battery or even off of 12 V car power. Car voltage is more like 13.6 when the engine is running, but that's still OK. Don't go higher though.
The 12DT8 information refers you to the 12AT7, for which there is considerably more detail:
This extra stuff doesn't matter if you just want to make the tube light up. Just connect 12 V between pins 4 and 5 and it should glow nicely after a few seconds.
It may be fun to see the tube light up, but it's a lot more fun to have it actually do something. Fortunately, this is a tube you can do some interesting things with yourself and see some results. For example, if you can find someone to help you with electronics a bit, you could wire this up to make a small amplifier. You can use a small loudspeaker as a microphone, and have the output of the amplifier drive a pair of headphones thru a audio transformer. Just be careful with the high voltage these things need. Again, have someone help you with this.