I have trouble with my chain popping off while riding pretty often, and that's even right out of the shop to get the chain tightened. (I'm not sure how to tighten a three speed hub.)
About once a week or so (I ride about 60-100 miles/week) i'll be riding and suddenly lose power and brakes.
I ride entirely in the road, but the roads here have been described as being "more technical than some singletrack" to me.
What do I need to do, if anything? How often do chains drop normally?

Mine never drops off.

I'd offer advice, but you mentioned having a multi-speed hub. I'm not quite sure what would cause it since you don't have a derailer to adjust.

Never.

I used to drop the chain on my first-generation Langster; it happened because the nuts next to the adjustable cones on the axle deformed the aluminum dropouts, allowing the axle to slip forward no matter how tight the outer nuts were. I threw on a banjo axle adjuster I had left over from my first track bike (got it in 1966). Then I bought a Felt TK2, which has steel-reinforced rear dropout faces.

never

you should true your chainring.

I throw it in the garbage every spring and/or couple thousand miles.

if you never throw your chain, you aren't riding hard enough! EXTEME!!!

I'd suggest disconnecting the IGH and adjusting the chainline, then reset the cable tension for the IGH and you should be good to roll.

This shouldn't be happening, so something is wrong. First verify the the the chainline is correct - stand behind the bike and sight along the rear cog to the chainring and see that they line up properly. Second check that your chainring doesn't have a wobble or a bent tooth that could let the chain jump off. Turn the crank and watch the chainring teeth to see if anything looks misaligned.

The third possibility is that the chainring isn't perfectly concentric with the bottom bracket spindle and therefore the chain tension varies as it goes around. Again slowly turn the crank and see if the tightness of the chain varies. If this is the problem you can frequently make minor corrections by slightly loosening the bolts holding the chainring to the crank and tapping the ring to make it more concentric before tightening the bolts again.

I only 'throw my chain' when it breaks mid-climb and makes me look like a jackass.

I did that during a mountain bike race years ago on a really sketchy technical climb, not fun.

As for the OP's issue, I almost never throw chains on my FG, but as was said above check chainline, tension, etc.?

If it's an everyday occurrence, have one of the guys from your LBS take it for a spin and try and reproduce the problem.

Do you find this to consistently and reliably work? I don't, so I'm curious.
I take the 5 minutes to measure, especially since I don't have to do it that often.

not even once.

+1

Sounds like you need to check your chainline

since I've been riding fixed.... never

Just once after cut chainring bolts failed.

Why only three bolts Stronglight TA?

Otherwise, should not be dropping chains. LBS, yo.

you inspire me.

Neva

I have never thrown it on my FG.

Never.

you can get away with a pretty bad chainline, but a little chainring wobble can do a number.

I hear a bit of a aluminiummey strainey ticking noise from my cranks/BB area somewhere when I accelerate hard, would that be related? (I'd say it was a creak, but a creak is a sustained noise and this is more like a click that sounds like a creak.)

Your chain is falling off, every noise coming from the drivetrain is related. But nola does have some rough roads, not to mention the street car tracks. Has the ability to rattle a bike apart, piece by piece.

May want to crank the tension up a notch.