Reducing the weight of your bicycle may improve your climbing, as you won't have to exert as much energy to climb, but there's more to it than that. The energy to climb a hill is dependent on the sum of your kinetic, potential energy and loss of mechanical energy. Mechanical energy can be a substantial loss in the energy system. Mechanical energy is highly dependent on the efficiency of your bicycle systems major structural and rotational components. The structural components are made up of the frame and the wheels. The rotational components consist of the elements of the drive chain, i.e., wheels, hubs, your shoes, crank and chain. If the structural components have excessive flex (titanium frames and wheels with few spokes) you are wasting energy flexing these components. If you have to spin heavy rotational components, you are exerting more energy than is needed. Lighter is the key to any component selection. However, for the structural members that can flex, stiffer and lighter are opposites. It is very difficult to have a structure, which is both very light and very stiff. It comes down to a very fine balance of these two properties.

Table 3 is a comparison of some popular carbon fiber and aerodynamic bicycles on the market. The weight comparison is straightforward. The aerodynamics comparison is a generalization, but the point is the frames, which offer aerodynamic benefits typically weight a bit more. One could also assume that the frames lateral stiffness would also follow the aerodynamic trend.

Yes, reduced weight or mass is a good thing to conserving rotational energy. In particular the selections of your aerodynamic wheels should consider where the mass is in the wheel design. The moment of rotational inertia is a function of the radius squared. If the rim is "heavy" or if the tires are "heavy " it will take more rotational energy to accelerate the wheel up to speed. This is part of reasons tandems don't accelerate very well, because the wheels are designed to support the weight and loads of two riders.

Spring energy represents the stiffness of your structural components and their ability to transfer energy to moving the bicycle forward. Your frame and wheels are the main elements in this equation. The energy lost is a squared function of the deflection distance. It is an energy loss because the deflection is perpendicular to the direction of motion and does not contribute to going forward. This energy loss occurs every time you turn the pedals one revolution. In general, this energy loss is small, but over long distances why lose it. Before the advent of composite wheels we have examples of sprinters trying to improve spring energy loss by stiffing their wheels by soldering the spokes together where they cross. If your race comes down to a sprint you want even pedal stroke to go towards getting you to the line first, not flexing the wheels and the frame sideways.