Should the bike or the rider take the jolt from a bump, and how? The last part of my commute to work has some significant bumps in the road which are really quite jarring, since I'm on a road bike that, of course, has no shocks (but has a carbon fork).

Most of the bumps are caused by cracks in the asphalt that have separated or curb corners that don't totally flatten to the road level at the crosswalk... or tree roots that have started to push up from beneath the pavement.

Should the bike take most of the concussion from hitting bumps? If I lift myself off the seat only barely and just for a moment, I don't have to feel the brunt of the force, and my bike has more freedom to take the hit instead of me. (These aren't so large that I am at much risk of totally flying off the bike if I stand up slightly for those moments. I'm still holding on, but maybe more lightly.)

Will the bike last longer if I kind of use my legs as springs, or should I stay firmly planted and take the jolt from the bump? What's the tradeoff, if any, between my health/safety and the longevity of the bike?
This is not an either-or proposition. Your bike is hitting the bumps and supporting your full weight (minus the very small proportion of weight that might be falling at that exact moment) regardless of how you stand when you hit the bumps.

The difference is whether you're going to let the additional damping effects of the down tube, seat tube, bottom bracket, cranks, and pedals absorb some of the shock before hitting your body, and whether you take that shock on your feet (which are protected by hard soles and socks and are connected to your springy ankles) or just take it right on the sit bones (which are hard and part of your skeleton). Obviously one of those is going to feel a lot worse than the other.

It's helpful to imagine an analogous system: a very light car carrying a heavy load. Our choice is whether the load is supported on a suspension system or not. If we rigidly attach the load to the frame of the car, the load isn't "taking the hit" when the car hits a bump -- rather, the shock is worsened because there's nothing to absorb some of the energy of the shock. If we suspend the load, the total weight the car is carrying remains the same (it has to), but we let the suspension system absorb some of the energy.

You and your bike are both better off if you get off the seat a bit and let your legs (and the frame, to some extent) act as a suspension system.