Should Subwoofer Be Plugged Into Same Circuit?

Help prevent, yes. Absolutely required, no.

It's pretty easy to tell when you hear it. :)

Run one line, 10 ga copper if you can. Not the cheaper aluminum. One run only, direct from panel to the one system outlet. Plug your power conditioner into it, and plug everything into the conditioner. Or power strip if you're not using a conditioner. 

Hum happens when there's more than one path to ground. But its unpredictable. You could plug into ten different circuits and have no problem, or you could use two and have intolerable hum. Why risk it? One and done. 

This is btw based on actual experience having run several circuits myself and heard the effect of different methods first hand. Also this will be enough no matter how many subs or watts you add down the road. Because contrary to what most probably think the greatest power load is not when the volume is cranked to 11 but the instant things are turned on and drawing full load to fill the power supply caps.

Bit of a shame going to all this trouble for multi-channel. Someday though you may decide to go for sound quality, put a stereo rig in there, and then it will all have been worth it.

It depends on the total current draw of all your components on a particular circuit. Calculate the total amperage that will be used at one time on the circuit. 
And it's best to use a 20A dedicated line which includes 20A wall receptacle and 20A breaker.

I’m in the USA by the way. I get the feeling MillerCarbon May be in another country since we haven’t used aluminum in decades this side of the pond, and I have never heard of using 10 gauge wire for a regular electric run (either 12 or 14).

Thats a good idea on making a 20 amp dedicated circuit for everything if there is even a chance it could increase chances of a ground loop. 
Don’t worry, I have a home theater and a stereo setup, I’m just working on them both at the same time. 
Also, I didn’t think you were supposed to plug everything into a power conditioner. I’m probably wrong because I have never had one but will most likely buy that Brick thinga-ma-bob.

It should be a law of audiophiles that everything should be plugged into one outlet. That is the best way to eliminate ground loops. And sometimes the only way.