So I've only been on a road bike for a week now and one odd thing I've noticed is that the headset bearing has absolutely no tension on it. When I pick the bike up, the front wheel goes completely limp and flops to one side.

Now I would've just accepted this as normal but I happened to watch a coworker of mine walk his bike through the hallway yesterday with just his hand on the seat (high end carbon bike, didnt catch the brand). The bike maintained a perfectly straight line.

Out of curiosity I tried this in my driveway last night and the front wheel was all kinds of wobbly. So this has me wondering if a front wheel/fork should ideally be completely free moving, or if through adjustment or replacement of the headset should actually have some tension.

Just FYI, I have 23C tires on the bike which I suppose could contribute to this (vs a wider tire).

Loosen the stem bolts and then tighten the top cap bolt. Then retighten the stem bolts.

what type of tension should I be looking for? Or is this all personal preference?

Enough that it's not binding the bearings.

The bearings should be free moving but not so loose that you can feel movement at the headset when you rock the bike with the front brake applied.

I'll check again tonight, but I dont recall any play in the headset itself, rather it seems that the bearings roll almost too loosely.

This depends on the headset. If you have no slop in it--no movement in the headset itself, either back and forth or up and down--you are fine. I have had headsets that felt very loose like this and others that felt more tight.

I dont suppose there is a quantitative measurement associated with each headset to specify its friction/resistance so that I can find the tightest bearings possible before buying

d00d, srsly?!? Just hold the bike straight if you want to walk it straight...

Also, the reason a bicycle can balance on two wheels alone is because the front wheel can constantly microadjust to keep the bike balanced. In fact, an unridden bike will naturally turn it's front wheel to keep it balanced until the point it eventually falls over.

There is no advantage to having an overly tight headset. If your bike can't roll straight there is something else going on. No play or rattle in the headset with you on or off the bike means it's properly adjusted.

No idea. Not that I know of. Might want to ask in the mechanics forum. I just don't worry about it as long as there's no slop.