If I bring it out of the water I try to support it with my other hand under the belly for a picture. The best way to remove a hook is probably keeping it in a net and in the water. Keep as much of the fish in the water for as long as you can IMHO. Also when this fish in at least partially in the water, the water keeps the fish buoyant and reduces its weight. I don't know the proper terms, but just think as a kid in the pool you could lift other kids in the water, but when they got a large portion of their mass out of the water, they become too heavy to lift. My point is it is less hard on the fish to handle it by the jaw alone if it is in the water.

As far as getting a weight, you can keep it in the net and then subtract the weight of the net later. I have a net made of a silicone like substance that is supposed to be better on the fish than an old nylon net style.

Oh and I don't know if it helps but I don't like to push the fishes (tongue) area up. I know this is a common maneuver to keep the fish from moving, but I feel it is damaging. This is all just my opinion.

I know there is a movement in the fly fishing world called something like keep'em wet, or release'em wet. Where they promote never taking the fish out of the water at all, not for hook removal or for pictures.

Get out and fish.